# Load packages
import numpy as np # The 'numpy' package is needed for matrix operations and calculations
import pandas as pd # The 'pandas' package helps us to import and manage data
import math as math # Math package for data cleaning
from scipy import stats # Import scipy.package for confidence intervals
from sklearn.preprocessing import StandardScaler # Import the package for standardizing data
import D47calib as D47c # Import the package for treating clumped isotope data by Daëron and Vermeesch (2023; https://github.com/mdaeron/D47calib)
import matplotlib.pyplot as plt # The 'matplotlib' package contains tools needed to plot our data and results
from matplotlib.lines import Line2D
from matplotlib.patches import Rectangle # The 'Rectangle' function is used to add rectangles to our plots
import seaborn as sns # The 'seaborn' package is used to make our plots look nicer (e.g. enable heatmaps)
import warnings # The 'warnings' package is used to suppress warnings that might occur during the calculations
%matplotlib inline

# Load model SST data as prior and show data structure
PlioMIP_SST = pd.read_csv('PWP case/PlioMIP_SST.csv') # Load the data for this assignment into Python and in the Jupyter environment.
PlioMIP_SST.head()

# Load model SAT data as prior and show data structure
PlioMIP_SAT = pd.read_csv('PWP case/PlioMIP_SAT.csv') # Load the data for this assignment into Python and in the Jupyter environment.
PlioMIP_SAT.head()

# Load model SSS data as prior and show data structure
PlioMIP_SSS = pd.read_csv('PWP case/PlioMIP_SSS.csv')  # Load the data for this assignment into Python and in the Jupyter environment.
PlioMIP_SSS = PlioMIP_SSS.pivot(index='modelname', columns='month', values='SSS').reset_index() # Pivot the data to have months as columns and models as rows
# Rename months to two-letter codes
month_mapping = {
    'Jan': 'ja', 'Feb': 'fb', 'Mar': 'mr', 'Apr': 'ar', 'May': 'my', 
    'Jun': 'jn', 'Jul': 'jl', 'Aug': 'ag', 'Sep': 'sp', 'Oct': 'ot', 
    'Nov': 'nv', 'Dec': 'dc'
}
PlioMIP_SSS.rename(columns=month_mapping, inplace=True) # Rename the columns to two-letter codes for months
PlioMIP_SSS.head()

# Load model precipitation data as prior and show data structure
PlioMIP_precip = pd.read_csv('PWP case/PlioMIP_precip.csv')  # Load the data for this assignment into Python and in the Jupyter environment.
PlioMIP_precip = PlioMIP_precip.pivot(index='modelname', columns='month', values='Mean_Precipitation').reset_index()  # Pivot the data to have months as columns and models as rows
PlioMIP_precip.rename(columns=month_mapping, inplace=True)  # Rename the columns to two-letter codes for months
PlioMIP_precip.head()

# Normalize model names to lowercase and remove "-" signs
def normalize_modelname(df):
    df['modelname'] = df['modelname'].str.lower().str.replace('-', '', regex=False)
    # Merge all model names starting with "ipslcm6" into one modelname
    df['modelname'] = df['modelname'].str.replace(r'^ipslcm6.*', 'ipslcm6', regex=True)
    return df

# Apply normalization to all datasets
PlioMIP_SST = normalize_modelname(PlioMIP_SST)
PlioMIP_SAT = normalize_modelname(PlioMIP_SAT)
PlioMIP_SSS = normalize_modelname(PlioMIP_SSS)
PlioMIP_precip = normalize_modelname(PlioMIP_precip)

# Merge the datasets of SST, SAT, SSS, and precipitation, force suffixes to be added to the column names
PlioMIP = pd.merge(
    PlioMIP_SST.rename(columns={c: c + '_SST' for c in PlioMIP_SST.columns if c != 'modelname'}),
    PlioMIP_SAT.rename(columns={c: c + '_SAT' for c in PlioMIP_SAT.columns if c != 'modelname'}),
    on='modelname',
    how='outer'
)

PlioMIP = pd.merge(
    PlioMIP,
    PlioMIP_SSS.rename(columns={c: c + '_SSS' for c in PlioMIP_SSS.columns if c != 'modelname'}),
    on='modelname',
    how='outer'
)

PlioMIP = pd.merge(
    PlioMIP,
    PlioMIP_precip.rename(columns={c: c + '_precip' for c in PlioMIP_precip.columns if c != 'modelname'}),
    on='modelname',
    how='outer'
)

# Display the combined dataset
PlioMIP.head()

# Create list of month names
months = ['ja', 'fb', 'mr', 'ar', 'my', 'jn', 'jl', 'ag', 'sp', 'ot', 'nv', 'dc']

# Prior SST, SAT, SSS & precipitation estimates from climate models (mean)
mu_prior_SAT_monthly = np.array(PlioMIP[[f"{month}_SAT" for month in months]].mean(axis=0, skipna=True))
mu_prior_SST_monthly = np.array(PlioMIP[[f"{month}_SST" for month in months]].mean(axis=0, skipna=True))
mu_prior_SSS_monthly = np.array(PlioMIP[[f"{month}_SSS" for month in months]].mean(axis=0, skipna=True))
mu_prior_precip_monthly = np.array(PlioMIP[[f"{month}_precip" for month in months]].mean(axis=0, skipna=True))

# Covariance between months in prior SST, SAT, SSS, and precip estimates from climate models (covariance matrix)
cov_prior_SAT_monthly = np.cov(PlioMIP[[f"{month}_SAT" for month in months]].dropna(), rowvar=False)
cov_prior_SST_monthly = np.cov(PlioMIP[[f"{month}_SST" for month in months]].dropna(), rowvar=False)
cov_prior_SSS_monthly = np.cov(PlioMIP[[f"{month}_SSS" for month in months]].dropna(), rowvar=False)
cov_prior_precip_monthly = np.cov(PlioMIP[[f"{month}_precip" for month in months]].dropna(), rowvar=False)

# Store copy of original prior means to keep when later updating the prior
mu_prior_SAT_monthly_original, cov_prior_SAT_monthly_original = mu_prior_SAT_monthly.copy(), cov_prior_SAT_monthly.copy()
mu_prior_SST_monthly_original, cov_prior_SST_monthly_original = mu_prior_SST_monthly.copy(), cov_prior_SST_monthly.copy()
mu_prior_SSS_monthly_original, cov_prior_SSS_monthly_original = mu_prior_SSS_monthly.copy(), cov_prior_SSS_monthly.copy()
mu_prior_precip_monthly_original, cov_prior_precip_monthly_original = mu_prior_precip_monthly.copy(), cov_prior_precip_monthly.copy()

# Extract the standard deviations (uncertainty) from the covariance matrix
std_prior_SAT_monthly = np.sqrt(np.diag(cov_prior_SAT_monthly))
std_prior_SST_monthly = np.sqrt(np.diag(cov_prior_SST_monthly))
std_prior_SSS_monthly = np.sqrt(np.diag(cov_prior_SSS_monthly))
std_prior_precip_monthly = np.sqrt(np.diag(cov_prior_precip_monthly))

print("SAT Monthly Means:", mu_prior_SAT_monthly)
print("SAT Monthly Std Devs:", std_prior_SAT_monthly)
print("SST Monthly Means:", mu_prior_SST_monthly)
print("SST Monthly Std Devs:", std_prior_SST_monthly)
print("SSS Monthly Means:", mu_prior_SSS_monthly)
print("SSS Monthly Std Devs:", std_prior_SSS_monthly)
print("Precip Monthly Means:", mu_prior_precip_monthly)
print("Precip Monthly Std Devs:", std_prior_precip_monthly)

SAT Monthly Means: [ 6.71411765  6.69294118  8.04647059 10.80058824 14.58529412 19.27941176
 22.28411765 21.87705882 18.43058824 14.13235294 10.31941176  7.91294118]
SAT Monthly Std Devs: [2.19619574 2.33777342 2.13652153 1.88298457 1.96586214 2.24523793
 2.7378734  2.89253514 2.75413206 2.61063098 2.36348384 2.20706469]
SST Monthly Means: [ 8.89909091  7.84181818  7.89181818  9.60909091 12.97272727 17.37909091
 21.1        22.14727273 20.38636364 17.25272727 13.83181818 10.91818182]
SST Monthly Std Devs: [1.66519341 1.70563078 1.62979028 1.41943971 1.74026487 2.5002778
 3.07210677 3.32125606 3.31004916 2.94745684 2.34808781 1.8649387 ]
SSS Monthly Means: [29.29601903 29.27455803 29.08572385 28.76846642 28.43459126 28.18546307
 28.01587662 27.98742991 28.17716438 28.56415277 28.93190974 29.18675145]
SSS Monthly Std Devs: [5.92900181 5.93098095 5.9695257  6.03338574 6.09152224 6.17221843
 6.2365113  6.23680216 6.19993142 6.09896377 6.00078604 5.9423462 ]
Precip Monthly Means: [3.35899606 2.79623688 2.18241293 1.75411782 1.52990136 1.37490782
 1.44029882 2.14277374 2.72446826 3.26583779 3.82762136 3.79819106]
Precip Monthly Std Devs: [0.43419051 0.4450087  0.45465416 0.31157176 0.20340219 0.28421396
 0.4935758  0.5372476  0.459027   0.3647094  0.31141638 0.5035945 ]

# Set dimensions of data
n_models_monthly = len(PlioMIP["modelname"])  # Find the total number of models (use monthly data because monthly data has this column duplicated 3 times)

# Create a monthly scale for the x-axis
month_names = ['January', 'February', 'March', 'April', 'May', 'June', 'July', 'August', 'September', 'October', 'November', 'December']  # List full month names
months_scale = np.arange(len(months)) + 1  # Create monthly scale

# Create the figure and axes
fig, axes = plt.subplots(2, 1, figsize=(10, 12), sharex=True)

# Panel 1: Plot the prior distribution for SST and SAT
axes[0].plot(months_scale, mu_prior_SAT_monthly, label='Prior SAT Mean', marker='o', color='r')
axes[0].plot(months_scale, mu_prior_SST_monthly, label='Prior SST Mean', marker='o', color='b')

# Add 95% confidence intervals for SAT
axes[0].fill_between(
    months_scale,
    mu_prior_SAT_monthly - stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_SAT_monthly / np.sqrt(n_models_monthly),
    mu_prior_SAT_monthly + stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_SAT_monthly / np.sqrt(n_models_monthly),
    alpha=0.2, color='r', label='SAT 95% CI'
)

# Add 95% confidence intervals for SST
axes[0].fill_between(
    months_scale,
    mu_prior_SST_monthly - stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_SST_monthly / np.sqrt(n_models_monthly),
    mu_prior_SST_monthly + stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_SST_monthly / np.sqrt(n_models_monthly),
    alpha=0.2, color='b', label='SST 95% CI'
)

axes[0].set_title('Prior Mean and 95% Confidence Interval for Monthly SST & SAT Values')
axes[0].set_ylabel('Temperature (°C)')
axes[0].legend()
axes[0].grid(True)

# Panel 2: Plot the prior distribution for SSS and precipitation
axes[1].plot(months_scale, mu_prior_SSS_monthly, label='Prior SSS Mean', marker='o', color='g')
ax2 = axes[1].twinx()  # Create a secondary y-axis for precipitation
ax2.plot(months_scale, mu_prior_precip_monthly, label='Prior Precipitation Mean', marker='o', color='purple')

# Add 95% confidence intervals for SSS
axes[1].fill_between(
    months_scale,
    mu_prior_SSS_monthly - stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_SSS_monthly / np.sqrt(n_models_monthly),
    mu_prior_SSS_monthly + stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_SSS_monthly / np.sqrt(n_models_monthly),
    alpha=0.2, color='g', label='SSS 95% CI'
)

# Add 95% confidence intervals for precipitation
ax2.fill_between(
    months_scale,
    mu_prior_precip_monthly - stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_precip_monthly / np.sqrt(n_models_monthly),
    mu_prior_precip_monthly + stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_precip_monthly / np.sqrt(n_models_monthly),
    alpha=0.2, color='purple', label='Precipitation 95% CI'
)

axes[1].set_ylabel('SSS (psu)', color='g')
ax2.set_ylabel('Precipitation (mm/day)', color='purple')
axes[1].set_title('Prior Mean and 95% Confidence Interval for Monthly SSS & Precipitation Values')
axes[1].legend(loc='upper left')
ax2.legend(loc='upper right')
axes[1].grid(True)

# Update the x-axis with month names
axes[1].set_xticks(months_scale)
axes[1].set_xticklabels(month_names, rotation=45, ha="right")

[Text(1, 0, 'January'),
 Text(2, 0, 'February'),
 Text(3, 0, 'March'),
 Text(4, 0, 'April'),
 Text(5, 0, 'May'),
 Text(6, 0, 'June'),
 Text(7, 0, 'July'),
 Text(8, 0, 'August'),
 Text(9, 0, 'September'),
 Text(10, 0, 'October'),
 Text(11, 0, 'November'),
 Text(12, 0, 'December')]

# Apply T47()-function from the D47calib package to all SST columns
# Identify the SST and SAT columns separately
SST_columns = [col for col in PlioMIP.columns if col.endswith('_SST')]
SAT_columns = [col for col in PlioMIP.columns if col.endswith('_SAT')]

# Apply the conversion function to the SST and SAT columns and add new columns for D47 and D47_SE
for col in SST_columns:
    base_col_name = col.replace('_SST', '') # Remove the '_SST' suffix from the column name
    PlioMIP[f'{base_col_name}_SST_D47'], PlioMIP[f'{base_col_name}_SST_D47_SE'] = zip(*PlioMIP[col].apply(
        lambda x: D47c.OGLS23.T47(T = x) if not pd.isna(x) else (np.nan, np.nan)
    )) # Use zip() to unpack the tuple returned by the apply() method and apply the T47()-function to each value in the column
for col in SAT_columns:
    base_col_name = col.replace('_SAT', '') # Remove the '_SST' suffix from the column name
    PlioMIP[f'{base_col_name}_SAT_D47'], PlioMIP[f'{base_col_name}_SAT_D47_SE'] = zip(*PlioMIP[col].apply(
        lambda x: D47c.OGLS23.T47(T = x) if not pd.isna(x) else (np.nan, np.nan)
    )) # Use zip() to unpack the tuple returned by the apply() method and apply the T47()-function to each value in the column

# Display the combined data with D47 and D47_SE columns
D47_columns = [col for col in PlioMIP.columns if col.endswith('_D47')]
D47_se_columns = [col for col in PlioMIP.columns if '_D47_SE' in col]
print("D47 values for all model outcomes:\n", PlioMIP[D47_columns].head())
print("Calibration standard errors for all model outcomes:\n", PlioMIP[D47_se_columns].head())

D47 values for all model outcomes:
    ja_SST_D47  fb_SST_D47  mr_SST_D47  ar_SST_D47  my_SST_D47  jn_SST_D47  \
0         NaN         NaN         NaN         NaN         NaN         NaN   
1    0.648316    0.652623    0.651496    0.643531    0.630811    0.613992   
2    0.642505    0.646808    0.646093    0.638746    0.626446    0.610876   
3         NaN         NaN         NaN         NaN         NaN         NaN   
4         NaN         NaN         NaN         NaN         NaN         NaN   

   jl_SST_D47  ag_SST_D47  sp_SST_D47  ot_SST_D47  ...  mr_SAT_D47  \
0         NaN         NaN         NaN         NaN  ...    0.643673   
1    0.601033    0.595364    0.600724    0.612897  ...    0.642505   
2    0.597955    0.592525    0.596581    0.606465  ...    0.639270   
3         NaN         NaN         NaN         NaN  ...    0.640392   
4         NaN         NaN         NaN         NaN  ...    0.636241   

   ar_SAT_D47  my_SAT_D47  jn_SAT_D47  jl_SAT_D47  ag_SAT_D47  sp_SAT_D47  \
0    0.635203    0.624669    0.610110    0.600076    0.599613    0.611997   
1    0.633515    0.620321    0.602833    0.592104    0.594243    0.609855   
2    0.631459    0.619170    0.605427    0.593338    0.593428    0.603238   
3    0.631733    0.619499    0.605993    0.596947    0.598139    0.608520   
4    0.629044    0.616294    0.601621    0.593338    0.592856    0.603238   

   ot_SAT_D47  nv_SAT_D47  dc_SAT_D47  
0    0.626177    0.638746    0.645451  
1    0.627455    0.642576    0.653061  
2    0.617696    0.632485    0.641834  
3    0.621974    0.632725    0.641024  
4    0.616424    0.628570    0.635203  

[5 rows x 24 columns]
Calibration standard errors for all model outcomes:
    ja_SST_D47_SE  fb_SST_D47_SE  mr_SST_D47_SE  ar_SST_D47_SE  my_SST_D47_SE  \
0            NaN            NaN            NaN            NaN            NaN   
1       0.001328       0.001362       0.001353       0.001292       0.001206   
2       0.001284       0.001316       0.001311       0.001258       0.001181   
3            NaN            NaN            NaN            NaN            NaN   
4            NaN            NaN            NaN            NaN            NaN   

   jn_SST_D47_SE  jl_SST_D47_SE  ag_SST_D47_SE  sp_SST_D47_SE  ot_SST_D47_SE  \
0            NaN            NaN            NaN            NaN            NaN   
1       0.001119       0.001073       0.001059       0.001072       0.001114   
2       0.001106       0.001065       0.001054       0.001062       0.001090   
3            NaN            NaN            NaN            NaN            NaN   
4            NaN            NaN            NaN            NaN            NaN   

   ...  mr_SAT_D47_SE  ar_SAT_D47_SE  my_SAT_D47_SE  jn_SAT_D47_SE  \
0  ...       0.001293       0.001234       0.001171       0.001103   
1  ...       0.001284       0.001223       0.001148       0.001078   
2  ...       0.001262       0.001210       0.001142       0.001086   
3  ...       0.001269       0.001212       0.001144       0.001088   
4  ...       0.001241       0.001196       0.001129       0.001074   

   jl_SAT_D47_SE  ag_SAT_D47_SE  sp_SAT_D47_SE  ot_SAT_D47_SE  nv_SAT_D47_SE  \
0       0.001070       0.001069       0.001110       0.001179       0.001258   
1       0.001053       0.001057       0.001102       0.001187       0.001285   
2       0.001055       0.001055       0.001079       0.001135       0.001217   
3       0.001062       0.001065       0.001097       0.001157       0.001218   
4       0.001055       0.001054       0.001079       0.001129       0.001193   

   dc_SAT_D47_SE  
0       0.001306  
1       0.001365  
2       0.001280  
3       0.001274  
4       0.001234  

[5 rows x 24 columns]

# Apply the d18Ow-SSS function from Harwood et al. (2007) to all SSS columns
# Identify the SSS columns
SSS_columns = [col for col in PlioMIP.columns if col.endswith('_SSS')]

# Apply the conversion function to the SSS columns and add new columns for d18Ow and d18Ow_SE
for col in SSS_columns:
    base_col_name = col.replace('_SSS', '')  # Remove the '_SSS' suffix from the column name
    PlioMIP[f'{base_col_name}_SSS_d18Ow_Harwood'] = PlioMIP[col].apply(
        lambda x: -9.300 + 0.274 * x if not pd.isna(x) else np.nan  # Calculate d18Ow using Harwood et al. (2007) equation
    )
    PlioMIP[f'{base_col_name}_SSS_d18Ow_Graniero'] = PlioMIP[col].apply(
        lambda x: -7.89 + 0.23 * x if not pd.isna(x) else np.nan  # Calculate d18Ow using Graniero et al. (2006) equation
    )
    PlioMIP[f'{base_col_name}_SSS_d18Ow_Railsback_ice'] = PlioMIP[col].apply(
        lambda x: 0 + 21 / 35 * (x - 35) if not pd.isna(x) else np.nan  # Calculate d18Ow using Railsback et al. (1989) equation with alpha for icehouse conditions
    )
    PlioMIP[f'{base_col_name}_SSS_d18Ow_Railsback_green'] = PlioMIP[col].apply(
        lambda x: -1.08 + 8 / 35 * (x - 35) if not pd.isna(x) else np.nan  # Calculate d18Ow using Railsback et al. (1989) equation with alpha for greenhouse conditions
    )

# Display the combined data with d18Ow and d18Ow_SE columns
d18Ow_columns = [col for col in PlioMIP.columns if "d18Ow" in col]
print("d18Ow values for all model outcomes:\n", PlioMIP[d18Ow_columns].head())

d18Ow values for all model outcomes:
    ar_SSS_d18Ow_Harwood  ar_SSS_d18Ow_Graniero  ar_SSS_d18Ow_Railsback_ice  \
0                   NaN                    NaN                         NaN   
1             -4.192196              -3.602427                   -9.815028   
2                   NaN                    NaN                         NaN   
3                   NaN                    NaN                         NaN   
4                   NaN                    NaN                         NaN   

   ar_SSS_d18Ow_Railsback_green  ag_SSS_d18Ow_Harwood  ag_SSS_d18Ow_Graniero  \
0                           NaN                   NaN                    NaN   
1                     -4.819058             -4.459904              -3.827146   
2                           NaN                   NaN                    NaN   
3                           NaN                   NaN                    NaN   
4                           NaN                   NaN                    NaN   

   ag_SSS_d18Ow_Railsback_ice  ag_SSS_d18Ow_Railsback_green  \
0                         NaN                           NaN   
1                   -10.40125                     -5.042381   
2                         NaN                           NaN   
3                         NaN                           NaN   
4                         NaN                           NaN   

   dc_SSS_d18Ow_Harwood  dc_SSS_d18Ow_Graniero  ...  \
0                   NaN                    NaN  ...   
1             -4.046049              -3.479749  ...   
2                   NaN                    NaN  ...   
3                   NaN                    NaN  ...   
4                   NaN                    NaN  ...   

   nv_SSS_d18Ow_Railsback_ice  nv_SSS_d18Ow_Railsback_green  \
0                         NaN                           NaN   
1                     -9.6828                     -4.768686   
2                         NaN                           NaN   
3                         NaN                           NaN   
4                         NaN                           NaN   

   ot_SSS_d18Ow_Harwood  ot_SSS_d18Ow_Graniero  ot_SSS_d18Ow_Railsback_ice  \
0                   NaN                    NaN                         NaN   
1             -4.243902               -3.64583                   -9.928252   
2                   NaN                    NaN                         NaN   
3                   NaN                    NaN                         NaN   
4                   NaN                    NaN                         NaN   

   ot_SSS_d18Ow_Railsback_green  sp_SSS_d18Ow_Harwood  sp_SSS_d18Ow_Graniero  \
0                           NaN                   NaN                    NaN   
1                     -4.862191             -4.385957              -3.765074   
2                           NaN                   NaN                    NaN   
3                           NaN                   NaN                    NaN   
4                           NaN                   NaN                    NaN   

   sp_SSS_d18Ow_Railsback_ice  sp_SSS_d18Ow_Railsback_green  
0                         NaN                           NaN  
1                  -10.239323                     -4.980694  
2                         NaN                           NaN  
3                         NaN                           NaN  
4                         NaN                           NaN  

[5 rows x 48 columns]

# Iterate over each model and calculate d18Oc values
for index, row in PlioMIP.iterrows():
    # Iterate over each month
    for month in months:
        SST = row[f"{month}_SST"]
        # Calculate d18Oc using different d18Ow-SSS relationships
        d18Ow_Harwood = row[f"{month}_SSS_d18Ow_Harwood"]
        d18Ow_Graniero = row[f"{month}_SSS_d18Ow_Graniero"]
        d18Ow_Railsback_ice = row[f"{month}_SSS_d18Ow_Railsback_ice"]
        d18Ow_Railsback_green = row[f"{month}_SSS_d18Ow_Railsback_green"]
        if not pd.isna(SST) and not pd.isna(d18Ow_Harwood):
            d18Oc_Harwood = (20.6 - SST) / 4.34 + (d18Ow_Harwood - 0.27)
        else:
            d18Oc_Harwood = np.nan
        if not pd.isna(SST) and not pd.isna(d18Ow_Graniero):
            d18Oc_Graniero = (20.6 - SST) / 4.34 + (d18Ow_Graniero - 0.27)
        else:
            d18Oc_Graniero = np.nan
        if not pd.isna(SST) and not pd.isna(d18Ow_Railsback_ice):
            d18Oc_Railsback_ice = (20.6 - SST) / 4.34 + (d18Ow_Railsback_ice - 0.27)
        else:
            d18Oc_Railsback_ice = np.nan
        if not pd.isna(SST) and not pd.isna(d18Ow_Railsback_green):
            d18Oc_Railsback_green = (20.6 - SST) / 4.34 + (d18Ow_Railsback_green - 0.27)
        else:
            d18Oc_Railsback_green = np.nan
        # Add the calculated d18Oc values to the DataFrame
        PlioMIP.loc[index, f"{month}_d18Oc_Harwood"] = d18Oc_Harwood
        PlioMIP.loc[index, f"{month}_d18Oc_Graniero"] = d18Oc_Graniero
        PlioMIP.loc[index, f"{month}_d18Oc_Railsback_ice"] = d18Oc_Railsback_ice
        PlioMIP.loc[index, f"{month}_d18Oc_Railsback_green"] = d18Oc_Railsback_green

# Display the updated DataFrame
PlioMIP.head()

C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:29: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Graniero"] = d18Oc_Graniero
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:30: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Railsback_ice"] = d18Oc_Railsback_ice
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:31: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Railsback_green"] = d18Oc_Railsback_green
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:28: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Harwood"] = d18Oc_Harwood
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:29: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Graniero"] = d18Oc_Graniero
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:30: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Railsback_ice"] = d18Oc_Railsback_ice
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:31: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Railsback_green"] = d18Oc_Railsback_green
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:28: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Harwood"] = d18Oc_Harwood
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:29: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Graniero"] = d18Oc_Graniero
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:30: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Railsback_ice"] = d18Oc_Railsback_ice
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:31: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Railsback_green"] = d18Oc_Railsback_green
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:28: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Harwood"] = d18Oc_Harwood
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:29: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Graniero"] = d18Oc_Graniero
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:30: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Railsback_ice"] = d18Oc_Railsback_ice
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:31: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Railsback_green"] = d18Oc_Railsback_green
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:28: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Harwood"] = d18Oc_Harwood
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:29: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Graniero"] = d18Oc_Graniero
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:30: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Railsback_ice"] = d18Oc_Railsback_ice
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:31: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Railsback_green"] = d18Oc_Railsback_green
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:28: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Harwood"] = d18Oc_Harwood
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:29: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Graniero"] = d18Oc_Graniero
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:30: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Railsback_ice"] = d18Oc_Railsback_ice
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:31: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Railsback_green"] = d18Oc_Railsback_green
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:28: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Harwood"] = d18Oc_Harwood
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:29: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Graniero"] = d18Oc_Graniero
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:30: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Railsback_ice"] = d18Oc_Railsback_ice
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:31: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Railsback_green"] = d18Oc_Railsback_green
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:28: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Harwood"] = d18Oc_Harwood
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:29: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Graniero"] = d18Oc_Graniero
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:30: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Railsback_ice"] = d18Oc_Railsback_ice
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:31: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Railsback_green"] = d18Oc_Railsback_green
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:28: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Harwood"] = d18Oc_Harwood
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:29: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Graniero"] = d18Oc_Graniero
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:30: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Railsback_ice"] = d18Oc_Railsback_ice
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:31: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Railsback_green"] = d18Oc_Railsback_green
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:28: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Harwood"] = d18Oc_Harwood
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:29: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Graniero"] = d18Oc_Graniero
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:30: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Railsback_ice"] = d18Oc_Railsback_ice
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:31: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Railsback_green"] = d18Oc_Railsback_green
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:28: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Harwood"] = d18Oc_Harwood
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:29: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Graniero"] = d18Oc_Graniero
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:30: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Railsback_ice"] = d18Oc_Railsback_ice
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:31: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Railsback_green"] = d18Oc_Railsback_green
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:28: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Harwood"] = d18Oc_Harwood
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:29: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Graniero"] = d18Oc_Graniero
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:30: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Railsback_ice"] = d18Oc_Railsback_ice
C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2217022968.py:31: PerformanceWarning: DataFrame is highly fragmented.  This is usually the result of calling `frame.insert` many times, which has poor performance.  Consider joining all columns at once using pd.concat(axis=1) instead. To get a de-fragmented frame, use `newframe = frame.copy()`
  PlioMIP.loc[index, f"{month}_d18Oc_Railsback_green"] = d18Oc_Railsback_green

# Set the weights of the data based on the standard errors
weights_monthly_SST_D47 = 1 / PlioMIP[[f"{month}_SST_D47_SE" for month in months]] ** 2
weights_monthly_SAT_D47 = 1 / PlioMIP[[f"{month}_SAT_D47_SE" for month in months]] ** 2

# Change the column suffixes from "_D47_SE" to "_D47" in weights_monthly_SST_D47 to match the headers of the D47 matrix later for multiplication
weights_monthly_SST_D47.columns = [col.replace('_SST_D47_SE', '_SST_D47') for col in weights_monthly_SST_D47.columns]
weights_monthly_SAT_D47.columns = [col.replace('_SAT_D47_SE', '_SAT_D47') for col in weights_monthly_SAT_D47.columns]

# Prior D47 estimates from climate models (weighted mean)
mu_prior_SST_D47_monthly = np.array((PlioMIP[[f"{month}_SST_D47" for month in months]] * weights_monthly_SST_D47).sum(axis = 0, skipna = True) / weights_monthly_SST_D47.sum(axis = 0, skipna = True)) # Calculate weighted monthly mean D47 values and convert to numpy array
mu_prior_SAT_D47_monthly = np.array((PlioMIP[[f"{month}_SAT_D47" for month in months]] * weights_monthly_SAT_D47).sum(axis = 0, skipna = True) / weights_monthly_SAT_D47.sum(axis = 0, skipna = True)) # Calculate weighted monthly mean D47 values and convert to numpy array

# Calculate simple (unweighted) mean for monthly d18Ow values using different SSS-d18Ow relationships
mu_prior_SSS_d18Ow_Harwood_monthly = np.array(PlioMIP[[f"{month}_SSS_d18Ow_Harwood" for month in months]].mean(axis=0, skipna=True))
mu_prior_SSS_d18Ow_Graniero_monthly = np.array(PlioMIP[[f"{month}_SSS_d18Ow_Graniero" for month in months]].mean(axis=0, skipna=True))
mu_prior_SSS_d18Ow_Railsback_ice_monthly = np.array(PlioMIP[[f"{month}_SSS_d18Ow_Railsback_ice" for month in months]].mean(axis=0, skipna=True))
mu_prior_SSS_d18Ow_Railsback_green_monthly = np.array(PlioMIP[[f"{month}_SSS_d18Ow_Railsback_green" for month in months]].mean(axis=0, skipna=True))
mu_prior_d18Oc_Harwood_monthly = np.array(PlioMIP[[f"{month}_d18Oc_Harwood" for month in months]].mean(axis=0, skipna=True))
mu_prior_d18Oc_Graniero_monthly = np.array(PlioMIP[[f"{month}_d18Oc_Graniero" for month in months]].mean(axis=0, skipna=True))
mu_prior_d18Oc_Railsback_ice_monthly = np.array(PlioMIP[[f"{month}_d18Oc_Railsback_ice" for month in months]].mean(axis=0, skipna=True))
mu_prior_d18Oc_Railsback_green_monthly = np.array(PlioMIP[[f"{month}_d18Oc_Railsback_green" for month in months]].mean(axis=0, skipna=True))

# Decompose variance within and between model outcomes
# Compute variance across models
model_variances_SST = PlioMIP[[f"{month}_SST_D47" for month in months]].var(axis = 0, ddof = 1)
model_variances_SAT = PlioMIP[[f"{month}_SAT_D47" for month in months]].var(axis = 0, ddof = 1)
model_variances_d18Ow_Harwood = PlioMIP[[f"{month}_SSS_d18Ow_Harwood" for month in months]].var(axis = 0, ddof = 1)
model_variances_d18Ow_Graniero = PlioMIP[[f"{month}_SSS_d18Ow_Graniero" for month in months]].var(axis = 0, ddof = 1)
model_variances_d18Ow_Railsback_ice = PlioMIP[[f"{month}_SSS_d18Ow_Railsback_ice" for month in months]].var(axis = 0, ddof = 1)
model_variances_d18Ow_Railsback_green = PlioMIP[[f"{month}_SSS_d18Ow_Railsback_green" for month in months]].var(axis = 0, ddof = 1)
model_variances_d18Oc_Harwood = PlioMIP[[f"{month}_d18Oc_Harwood" for month in months]].var(axis = 0, ddof = 1)
model_variances_d18Oc_Graniero = PlioMIP[[f"{month}_d18Oc_Graniero" for month in months]].var(axis = 0, ddof = 1)
model_variances_d18Oc_Railsback_ice = PlioMIP[[f"{month}_d18Oc_Railsback_ice" for month in months]].var(axis = 0, ddof = 1)
model_variances_d18Oc_Railsback_green = PlioMIP[[f"{month}_d18Oc_Railsback_green" for month in months]].var(axis = 0, ddof = 1)
# Compute variance on measurements
measurement_variances_SST = (PlioMIP[[f"{month}_SST_D47_SE" for month in months]] ** 2).mean(axis = 0, skipna = True)
measurement_variances_SAT = (PlioMIP[[f"{month}_SAT_D47_SE" for month in months]] ** 2).mean(axis = 0, skipna = True)

# Covariance between months in prior D47 estimates from climate models (weighted covariance matrix)
# Compute the covariance matrix for the raw data (without measurement uncertainty)
cov_raw_monthly_SST = np.cov(PlioMIP[[f"{month}_SST_D47" for month in months]].dropna(), rowvar = False)
cov_raw_monthly_SAT = np.cov(PlioMIP[[f"{month}_SAT_D47" for month in months]].dropna(), rowvar = False)
cov_raw_monthly_d18Ow_Harwood = np.cov(PlioMIP[[f"{month}_SSS_d18Ow_Harwood" for month in months]].dropna(), rowvar = False)
cov_raw_monthly_d18Ow_Graniero = np.cov(PlioMIP[[f"{month}_SSS_d18Ow_Graniero" for month in months]].dropna(), rowvar = False)
cov_raw_monthly_d18Ow_Railsback_ice = np.cov(PlioMIP[[f"{month}_SSS_d18Ow_Railsback_ice" for month in months]].dropna(), rowvar = False)
cov_raw_monthly_d18Ow_Railsback_green = np.cov(PlioMIP[[f"{month}_SSS_d18Ow_Railsback_green" for month in months]].dropna(), rowvar = False)
cov_raw_monthly_d18Oc_Harwood = np.cov(PlioMIP[[f"{month}_d18Oc_Harwood" for month in months]].dropna(), rowvar = False)
cov_raw_monthly_d18Oc_Graniero = np.cov(PlioMIP[[f"{month}_d18Oc_Graniero" for month in months]].dropna(), rowvar = False)
cov_raw_monthly_d18Oc_Railsback_ice = np.cov(PlioMIP[[f"{month}_d18Oc_Railsback_ice" for month in months]].dropna(), rowvar = False)
cov_raw_monthly_d18Oc_Railsback_green = np.cov(PlioMIP[[f"{month}_d18Oc_Railsback_green" for month in months]].dropna(), rowvar = False)
cov_prior_SST_D47_monthly = cov_raw_monthly_SST.copy() # Copy covariance matrix to add uncertainty coming from the measurements
cov_prior_SAT_D47_monthly = cov_raw_monthly_SAT.copy() # Copy covariance matrix to add uncertainty coming from the measurements
np.fill_diagonal(cov_prior_SST_D47_monthly, np.diagonal(cov_raw_monthly_SST) + measurement_variances_SST)  # Add diagonal terms for measurement uncertainties (which have no covariance between models)
np.fill_diagonal(cov_prior_SAT_D47_monthly, np.diagonal(cov_raw_monthly_SAT) + measurement_variances_SAT)  # Add diagonal terms for measurement uncertainties (which have no covariance between models)

# Store copy of original prior means to keep when later updating the prior
mu_prior_SST_D47_monthly_original, cov_prior_SST_D47_monthly_original = mu_prior_SST_D47_monthly.copy(), cov_prior_SST_D47_monthly.copy()
mu_prior_SAT_D47_monthly_original, cov_prior_SAT_D47_monthly_original = mu_prior_SAT_D47_monthly.copy(), cov_prior_SAT_D47_monthly.copy()
mu_prior_SSS_d18Ow_Harwood_monthly_original, mu_prior_d18Oc_Graniero_monthly_original, mu_prior_d18Oc_Railsback_ice_monthly_original, mu_prior_d18Oc_Railsback_green_monthly_original = mu_prior_SSS_d18Ow_Harwood_monthly.copy(), mu_prior_d18Oc_Graniero_monthly.copy(), mu_prior_d18Oc_Railsback_ice_monthly.copy(), mu_prior_d18Oc_Railsback_green_monthly.copy()
cov_prior_SSS_d18Ow_Harwood_monthly_original, cov_prior_d18Ow_Graniero_monthly_original, cov_prior_d18Ow_Railsback_ice_monthly_original, cov_prior_d18Ow_Railsback_green_monthly_original = cov_raw_monthly_d18Ow_Harwood.copy(), cov_raw_monthly_d18Ow_Graniero.copy(), cov_raw_monthly_d18Ow_Railsback_ice.copy(), cov_raw_monthly_d18Ow_Railsback_green.copy()
mu_prior_d18Oc_Harwood_monthly_original, mu_prior_d18Oc_Graniero_monthly_original, mu_prior_d18Oc_Railsback_ice_monthly_original, mu_prior_d18Oc_Railsback_green_monthly_original = mu_prior_d18Oc_Harwood_monthly.copy(), mu_prior_d18Oc_Graniero_monthly.copy(), mu_prior_d18Oc_Railsback_ice_monthly.copy(), mu_prior_d18Oc_Railsback_green_monthly.copy()
cov_prior_d18Oc_Harwood_monthly_original, cov_prior_d18Oc_Graniero_monthly_original, cov_prior_d18Oc_Railsback_ice_monthly_original, cov_prior_d18Oc_Railsback_green_monthly_original = cov_raw_monthly_d18Oc_Harwood.copy(), cov_raw_monthly_d18Oc_Graniero.copy(), cov_raw_monthly_d18Oc_Railsback_ice.copy(), cov_raw_monthly_d18Oc_Railsback_green.copy()

# Extract the standard deviations (uncertainty) from the covariance matrix
std_prior_SST_D47_monthly = np.sqrt(np.diag(cov_prior_SST_D47_monthly))
std_prior_SAT_D47_monthly = np.sqrt(np.diag(cov_prior_SAT_D47_monthly))
std_prior_SSS_d18Ow_Harwood_monthly = np.sqrt(np.diag(cov_raw_monthly_d18Ow_Harwood))
std_prior_SSS_d18Ow_Graniero_monthly = np.sqrt(np.diag(cov_raw_monthly_d18Ow_Graniero))
std_prior_SSS_d18Ow_Railsback_ice_monthly = np.sqrt(np.diag(cov_raw_monthly_d18Ow_Railsback_ice))
std_prior_SSS_d18Ow_Railsback_green_monthly = np.sqrt(np.diag(cov_raw_monthly_d18Ow_Railsback_green))
std_prior_d18Oc_Harwood_monthly = np.sqrt(np.diag(cov_raw_monthly_d18Oc_Harwood))
std_prior_d18Oc_Graniero_monthly = np.sqrt(np.diag(cov_raw_monthly_d18Oc_Graniero))
std_prior_d18Oc_Railsback_ice_monthly = np.sqrt(np.diag(cov_raw_monthly_d18Oc_Railsback_ice))
std_prior_d18Oc_Railsback_green_monthly = np.sqrt(np.diag(cov_raw_monthly_d18Oc_Railsback_green))

# Print the results
print("Prior D47 estimates from SST in climate models (weighted mean):")
print(mu_prior_SST_D47_monthly)
print("Prior D47 estimates from SST in climate models (weighted covariance matrix):")
print(std_prior_SST_D47_monthly)
print("---")
print("Prior D47 estimates from SAT in climate models (weighted mean):")
print(mu_prior_SAT_D47_monthly)
print("Prior D47 estimates from SAT in climate models (weighted covariance matrix):")
print(std_prior_SAT_D47_monthly)
print("---")
print("Prior d18Ow estimates from SSS in climate models (weighted mean) using Harwood et al.:")
print(mu_prior_SSS_d18Ow_Harwood_monthly)
print("Prior d18Ow estimates from SSS in climate models (weighted mean) using Graniero et al.:")
print(mu_prior_SSS_d18Ow_Graniero_monthly)
print("Prior d18Ow estimates from SSS in climate models (weighted mean) using Railsback et al. icehouse):")
print(mu_prior_SSS_d18Ow_Railsback_ice_monthly)
print("Prior d18Ow estimates from SSS in climate models (weighted mean) using Railsback et al. greenhouse):")
print(mu_prior_SSS_d18Ow_Railsback_green_monthly)
print("---")
print("Prior d18Ow estimates from SSS in climate models (weighted covariance matrix) using Harwood et al.:")
print(std_prior_SSS_d18Ow_Harwood_monthly)
print("Prior d18Ow estimates from SSS in climate models (weighted covariance matrix) using Graniero et al.:")
print(std_prior_SSS_d18Ow_Graniero_monthly)
print("Prior d18Ow estimates from SSS in climate models (weighted covariance matrix) using Railsback et al. icehouse):")
print(std_prior_SSS_d18Ow_Railsback_ice_monthly)
print("Prior d18Ow estimates from SSS in climate models (weighted covariance matrix) using Railsback et al. greenhouse):")
print(std_prior_SSS_d18Ow_Railsback_green_monthly)
print("---")
print("Prior d18Oc estimates from SST and d18Ow in climate models (weighted mean) using Harwood et al.:")
print(mu_prior_d18Oc_Harwood_monthly)
print("Prior d18Oc estimates from SST and d18Ow in climate models (weighted mean) using Graniero et al.:")
print(mu_prior_d18Oc_Graniero_monthly)
print("Prior d18Oc estimates from SST and d18Ow in climate models (weighted mean) using Railsback et al. icehouse):")
print(mu_prior_d18Oc_Railsback_ice_monthly)
print("Prior d18Oc estimates from SST and d18Ow in climate models (weighted mean) using Railsback et al. greenhouse):")
print(mu_prior_d18Oc_Railsback_green_monthly)
print("---")
print("Prior d18Oc estimates from SST and d18Ow in climate models (weighted covariance matrix) using Harwood et al.:")
print(std_prior_d18Oc_Harwood_monthly)
print("Prior d18Oc estimates from SST and d18Ow in climate models (weighted covariance matrix) using Graniero et al.:")
print(std_prior_d18Oc_Graniero_monthly)
print("Prior d18Oc estimates from SST and d18Ow in climate models (weighted covariance matrix) using Railsback et al. icehouse):")
print(std_prior_d18Oc_Railsback_ice_monthly)
print("Prior d18Oc estimates from SST and d18Ow in climate models (weighted covariance matrix) using Railsback et al. greenhouse):")
print(std_prior_d18Oc_Railsback_green_monthly)

Prior D47 estimates from SST in climate models (weighted mean):
[0.64595815 0.64972969 0.64957918 0.64352257 0.63174847 0.61687649
 0.60495042 0.60168179 0.60710259 0.61713861 0.62862875 0.63877671]
Prior D47 estimates from SST in climate models (weighted covariance matrix):
[0.00614572 0.00638619 0.00611021 0.00523351 0.00611762 0.0083541
 0.00986818 0.01053415 0.01065771 0.00977107 0.00805759 0.00665147]
---
Prior D47 estimates from SAT in climate models (weighted mean):
[0.6535531  0.65354873 0.64877444 0.63916405 0.62623172 0.61087967
 0.6014159  0.6026324  0.61343328 0.62749031 0.64061857 0.64921573]
Prior D47 estimates from SAT in climate models (weighted covariance matrix):
[0.0081692  0.00869841 0.00782808 0.00672115 0.00673335 0.00732744
 0.0086457  0.0091529  0.00900611 0.00891738 0.00842026 0.00808888]
---
Prior d18Ow estimates from SSS in climate models (weighted mean) using Harwood et al.:
[-1.27289079 -1.2787711  -1.33051166 -1.4174402  -1.50892199 -1.57718312
 -1.62364981 -1.6314442  -1.57945696 -1.47342214 -1.37265673 -1.3028301 ]
Prior d18Ow estimates from SSS in climate models (weighted mean) using Graniero et al.:
[-1.15191562 -1.15685165 -1.20028351 -1.27325272 -1.35004401 -1.40734349
 -1.44634838 -1.45289112 -1.40925219 -1.32024486 -1.23566076 -1.17704717]
Prior d18Ow estimates from SSS in climate models (weighted mean) using Railsback et al. icehouse):
[-3.42238858 -3.43526518 -3.54856569 -3.73892015 -3.93924524 -4.08872216
 -4.19047403 -4.20754205 -4.09370137 -3.86150834 -3.64085415 -3.48794913]
Prior d18Ow estimates from SSS in climate models (weighted mean) using Railsback et al. greenhouse):
[-2.38376708 -2.38867245 -2.43183455 -2.50435053 -2.58066485 -2.63760844
 -2.67637106 -2.68287316 -2.63950528 -2.55105079 -2.46699206 -2.40874252]
---
Prior d18Ow estimates from SSS in climate models (weighted covariance matrix) using Harwood et al.:
[1.6245465  1.62508878 1.63565004 1.65314769 1.66907709 1.69118785
 1.7088041  1.70888379 1.69878121 1.67111607 1.64421538 1.62820286]
Prior d18Ow estimates from SSS in climate models (weighted covariance matrix) using Graniero et al.:
[1.36367042 1.36412562 1.37299091 1.38767872 1.40105012 1.41961024
 1.4343976  1.4344645  1.42598423 1.40276167 1.38018079 1.36673963]
Prior d18Ow estimates from SSS in climate models (weighted covariance matrix) using Railsback et al. icehouse):
[3.55740109 3.55858857 3.58171542 3.62003145 3.65491335 3.70333106
 3.74190678 3.7420813  3.71995885 3.65937826 3.60047163 3.56540772]
Prior d18Ow estimates from SSS in climate models (weighted covariance matrix) using Railsback et al. greenhouse):
[1.35520041 1.35565279 1.36446302 1.3790596  1.39234794 1.41079278
 1.4254883  1.42555478 1.41712718 1.39404886 1.37160824 1.35825056]
---
Prior d18Oc estimates from SST and d18Ow in climate models (weighted mean) using Harwood et al.:
[ 1.24955356  1.48821276  1.44246505  0.99731593  0.16354539 -0.89974455
 -1.79850605 -2.05495028 -1.59818088 -0.78594795  0.05669023  0.76574859]
Prior d18Oc estimates from SST and d18Ow in climate models (weighted mean) using Graniero et al.:
[ 1.3438645   1.58320839  1.54290413  1.10686585  0.28345529 -0.77220851
 -1.66492102 -1.91919395 -1.46922561 -0.67007851  0.16162142  0.86311472]
Prior d18Oc estimates from SST and d18Ow in climate models (weighted mean) using Railsback et al. icehouse):
[-0.70238658 -0.46880033 -0.55487901 -1.06753118 -1.97805979 -3.09785252
 -4.04143149 -4.31396305 -3.80680401 -2.89761701 -1.97393634 -1.20882774]
Prior d18Oc estimates from SST and d18Ow in climate models (weighted mean) using Railsback et al. greenhouse):
[ 0.11114732  0.35051345  0.31038591 -0.12535656 -0.94843075 -2.00384695
 -2.89636307 -3.1505655  -2.70081797 -1.90209574 -1.07075095 -0.36950327]
---
Prior d18Oc estimates from SST and d18Ow in climate models (weighted covariance matrix) using Harwood et al.:
[1.81280103 1.77733509 1.77427535 1.7740801  1.73946053 1.7191968
 1.70980795 1.76480456 1.80080386 1.82777258 1.88551146 1.84844074]
Prior d18Oc estimates from SST and d18Ow in climate models (weighted covariance matrix) using Graniero et al.:
[1.50177344 1.46445261 1.4599008  1.45968833 1.42846825 1.41480843
 1.40968928 1.46695413 1.50185947 1.52472745 1.58016894 1.54079378]
Prior d18Oc estimates from SST and d18Ow in climate models (weighted covariance matrix) using Railsback et al. icehouse):
[4.13280493 4.10382652 4.10968154 4.11474429 4.07333031 4.04541763
 4.03417247 4.08110728 4.11199059 4.13652445 4.19181911 4.15737949]
Prior d18Oc estimates from SST and d18Ow in climate models (weighted covariance matrix) using Railsback et al. greenhouse):
[1.49169831 1.4543067  1.44970325 1.44949805 1.41841682 1.40503469
 1.40009942 1.45744644 1.49229456 1.51498084 1.57031808 1.53084828]

# Plot monthly prior distribution
fig, axes = plt.subplots(2, 2, figsize=(15, 12))  # Adjust the figure to have 2x2 grid

# Plot the prior distribution for SST
axes[0, 0].plot(months_scale, mu_prior_SST_D47_monthly, label='Prior SST Mean', color='b', marker='o')
axes[0, 0].fill_between(months_scale,
                        mu_prior_SST_D47_monthly - stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_SST_D47_monthly / np.sqrt(n_models_monthly),
                        mu_prior_SST_D47_monthly + stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_SST_D47_monthly / np.sqrt(n_models_monthly),
                        color='b', alpha=0.2, label='95% Confidence Interval')
axes[0, 0].set_xticks(months_scale)
axes[0, 0].set_xticklabels(month_names, rotation=45, ha="right")
axes[0, 0].set_title('Prior Mean and 95% Confidence Interval for Monthly SST D47 values')
axes[0, 0].set_xlabel('Month')
axes[0, 0].set_ylabel('D47 value')
axes[0, 0].legend()
axes[0, 0].grid(True)

# Plot the prior distribution for SAT
axes[0, 1].plot(months_scale, mu_prior_SAT_D47_monthly, label='Prior SAT Mean', color='r', marker='o')
axes[0, 1].fill_between(months_scale,
                        mu_prior_SAT_D47_monthly - stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_SAT_D47_monthly / np.sqrt(n_models_monthly),
                        mu_prior_SAT_D47_monthly + stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_SAT_D47_monthly / np.sqrt(n_models_monthly),
                        color='r', alpha=0.2, label='95% Confidence Interval')
axes[0, 1].set_xticks(months_scale)
axes[0, 1].set_xticklabels(month_names, rotation=45, ha="right")
axes[0, 1].set_title('Prior Mean and 95% Confidence Interval for Monthly SAT D47 values')
axes[0, 1].set_xlabel('Month')
axes[0, 1].set_ylabel('D47 value')
axes[0, 1].legend()
axes[0, 1].grid(True)

# Plot the prior distribution for d18Oc
axes[1, 0].plot(months_scale, mu_prior_d18Oc_Harwood_monthly, label='Prior d18Oc Mean', color='purple', marker='o')
axes[1, 0].fill_between(months_scale,
                        mu_prior_d18Oc_Harwood_monthly - stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_d18Oc_Harwood_monthly / np.sqrt(n_models_monthly),
                        mu_prior_d18Oc_Harwood_monthly + stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_d18Oc_Harwood_monthly / np.sqrt(n_models_monthly),
                        color='purple', alpha=0.2, label='95% Confidence Interval')
axes[1, 0].set_xticks(months_scale)
axes[1, 0].set_xticklabels(month_names, rotation=45, ha="right")
axes[1, 0].set_title('Prior Mean and 95% Confidence Interval for Monthly d18Oc values\n(based on Harwood et al. relationship)')
axes[1, 0].set_xlabel('Month')
axes[1, 0].set_ylabel('d18Oc value')
axes[1, 0].legend()
axes[1, 0].grid(True)

# Plot the prior distribution for precipitation
axes[1, 1].plot(months_scale, mu_prior_precip_monthly, label='Prior Precipitation Mean', color='teal', marker='o')
axes[1, 1].fill_between(months_scale,
                        mu_prior_precip_monthly - stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_precip_monthly / np.sqrt(n_models_monthly),
                        mu_prior_precip_monthly + stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_precip_monthly / np.sqrt(n_models_monthly),
                        color='teal', alpha=0.2, label='95% Confidence Interval')
axes[1, 1].set_xticks(months_scale)
axes[1, 1].set_xticklabels(month_names, rotation=45, ha="right")
axes[1, 1].set_title('Prior Mean and 95% Confidence Interval for Monthly Precipitation values')
axes[1, 1].set_xlabel('Month')
axes[1, 1].set_ylabel('Precipitation (mm/day)')
axes[1, 1].legend()
axes[1, 1].grid(True)

# Update the layout and show the plot
plt.tight_layout()
plt.show()

# Plot monthly prior distribution
fig, axes = plt.subplots(2, 2, figsize=(15, 12))  # Adjust the figure to have 2x2 grid

# Plot the prior distribution for d18Oc based on Harwood et al. (2007) relationship
axes[0, 0].plot(months_scale, mu_prior_d18Oc_Harwood_monthly, label='Prior d18Oc Mean', color='b', marker='o')
axes[0, 0].fill_between(months_scale,
                        mu_prior_d18Oc_Harwood_monthly - stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_d18Oc_Harwood_monthly / np.sqrt(n_models_monthly),
                        mu_prior_d18Oc_Harwood_monthly + stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_d18Oc_Harwood_monthly / np.sqrt(n_models_monthly),
                        color='b', alpha=0.2, label='95% Confidence Interval')
axes[0, 0].set_xticks(months_scale)
axes[0, 0].set_xticklabels(month_names, rotation=45, ha="right")
axes[0, 0].set_title('Prior Mean and 95% Confidence Interval for Monthly d18Oc values\nfollowing Harwood et al. relationship')
axes[0, 0].set_xlabel('Month')
axes[0, 0].set_ylabel('d18Oc value')
axes[0, 0].set_ylim([-6.5, 2.5])  # Set y-axis limits for better comparison between plots
axes[0, 0].legend()
axes[0, 0].grid(True)

# Plot the prior distribution for d18Oc based on Graniero et al. (2006) relationship
axes[0, 1].plot(months_scale, mu_prior_d18Oc_Graniero_monthly, label='Prior d18Oc Mean', color='r', marker='o')
axes[0, 1].fill_between(months_scale,
                        mu_prior_d18Oc_Graniero_monthly - stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_d18Oc_Graniero_monthly / np.sqrt(n_models_monthly),
                        mu_prior_d18Oc_Graniero_monthly + stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_d18Oc_Graniero_monthly / np.sqrt(n_models_monthly),
                        color='r', alpha=0.2, label='95% Confidence Interval')
axes[0, 1].set_xticks(months_scale)
axes[0, 1].set_xticklabels(month_names, rotation=45, ha="right")
axes[0, 1].set_title('Prior Mean and 95% Confidence Interval for Monthly d18Oc values\nfollowing Graniero et al. relationship')
axes[0, 1].set_xlabel('Month')
axes[0, 1].set_ylabel('d18Oc value')
axes[0, 1].set_ylim([-6.5, 2.5])  # Set y-axis limits for better comparison between plots
axes[0, 1].legend()
axes[0, 1].grid(True)

# Plot the prior distribution for d18Oc based on Railsback et al. (1989) icehouse relationship
axes[1, 0].plot(months_scale, mu_prior_d18Oc_Railsback_ice_monthly, label='Prior d18Oc Mean', color='purple', marker='o')
axes[1, 0].fill_between(months_scale,
                        mu_prior_d18Oc_Railsback_ice_monthly - stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_d18Oc_Railsback_ice_monthly / np.sqrt(n_models_monthly),
                        mu_prior_d18Oc_Railsback_ice_monthly + stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_d18Oc_Railsback_ice_monthly / np.sqrt(n_models_monthly),
                        color='purple', alpha=0.2, label='95% Confidence Interval')
axes[1, 0].set_xticks(months_scale)
axes[1, 0].set_xticklabels(month_names, rotation=45, ha="right")
axes[1, 0].set_title('Prior Mean and 95% Confidence Interval for Monthly d18Oc values\nfollowing Railsback et al. icehouse relationship')
axes[1, 0].set_xlabel('Month')
axes[1, 0].set_ylabel('d18Oc value')
axes[1, 0].set_ylim([-6.5, 2.5])  # Set y-axis limits for better comparison between plots
axes[1, 0].legend()
axes[1, 0].grid(True)

# Plot the prior distribution for d18Oc based on Railsback et al. (1989) greenhouse relationship
axes[1, 1].plot(months_scale, mu_prior_d18Oc_Railsback_green_monthly, label='Prior d18Oc Mean', color='teal', marker='o')
axes[1, 1].fill_between(months_scale,
                        mu_prior_d18Oc_Railsback_green_monthly - stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_d18Oc_Railsback_green_monthly / np.sqrt(n_models_monthly),
                        mu_prior_d18Oc_Railsback_green_monthly + stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_d18Oc_Railsback_green_monthly / np.sqrt(n_models_monthly),
                        color='teal', alpha=0.2, label='95% Confidence Interval')
axes[1, 1].set_xticks(months_scale)
axes[1, 1].set_xticklabels(month_names, rotation=45, ha="right")
axes[1, 1].set_title('Prior Mean and 95% Confidence Interval for Monthly d18Oc values\nfollowing Railsback et al. greenhouse relationship')
axes[1, 1].set_xlabel('Month')
axes[1, 1].set_ylabel('d18Oc value')
axes[1, 1].set_ylim([-6.5, 2.5])  # Set y-axis limits for better comparison between plots
axes[1, 1].legend()
axes[1, 1].grid(True)

# Update the layout and show the plot
plt.tight_layout()
plt.show()

# Define column names for SAT, SST, d18Oc, and precipitation
SAT_D47_columns_monthly = [f"{month}_SAT_D47" for month in months]
SST_D47_columns_monthly = [f"{month}_SST_D47" for month in months]
precip_columns_monthly = [f"{month}_precip" for month in months]
# Separate out d18Oc values calculated using different SSS-d18Ow relationships
d18Oc_Harwood_columns_monthly = [f"{month}_d18Oc_Harwood" for month in months]
d18Oc_Graniero_columns_monthly = [f"{month}_d18Oc_Graniero" for month in months]
d18Oc_Railsback_ice_columns_monthly = [f"{month}_d18Oc_Railsback_ice" for month in months]
d18Oc_Railsback_green_columns_monthly = [f"{month}_d18Oc_Railsback_green" for month in months]

# Combine the relevant columns into a single dataframe
combined_data_monthly_Harwood = PlioMIP[SAT_D47_columns_monthly + SST_D47_columns_monthly + d18Oc_Harwood_columns_monthly + precip_columns_monthly]
combined_data_monthly_Graniero = PlioMIP[SAT_D47_columns_monthly + SST_D47_columns_monthly + d18Oc_Graniero_columns_monthly + precip_columns_monthly]
combined_data_monthly_Railsback_ice = PlioMIP[SAT_D47_columns_monthly + SST_D47_columns_monthly + d18Oc_Railsback_ice_columns_monthly + precip_columns_monthly]
combined_data_monthly_Railsback_green = PlioMIP[SAT_D47_columns_monthly + SST_D47_columns_monthly + d18Oc_Railsback_green_columns_monthly + precip_columns_monthly]

# Calculate the covariance matrix for the combined data
cov_combined_monthly_Harwood = np.cov(combined_data_monthly_Harwood.dropna(), rowvar=False)
cov_combined_monthly_Graniero = np.cov(combined_data_monthly_Graniero.dropna(), rowvar=False)
cov_combined_monthly_Railsback_ice = np.cov(combined_data_monthly_Railsback_ice.dropna(), rowvar=False)
cov_combined_monthly_Railsback_green = np.cov(combined_data_monthly_Railsback_green.dropna(), rowvar=False)

# Plot the heatmap of the raw combined covariance matrix (following Harwood et al., as example)
plt.figure(figsize=(12, 10))
sns.heatmap(
    cov_combined_monthly_Harwood,  # Use the raw covariance matrix
    annot=False,
    fmt=".2f",
    cmap="coolwarm",
    center=0,
    xticklabels=SAT_D47_columns_monthly + SST_D47_columns_monthly + d18Oc_Harwood_columns_monthly + precip_columns_monthly,
    yticklabels=SAT_D47_columns_monthly + SST_D47_columns_monthly + d18Oc_Harwood_columns_monthly + precip_columns_monthly
)

# Add titles to the axes per parameter
plt.axvline(x=len(SAT_D47_columns_monthly), color='black', linestyle='--', linewidth=1)
plt.axvline(x=len(SAT_D47_columns_monthly) + len(SST_D47_columns_monthly), color='black', linestyle='--', linewidth=1)
plt.axvline(x=len(SAT_D47_columns_monthly) + len(SST_D47_columns_monthly) + len(d18Oc_Harwood_columns_monthly), color='black', linestyle='--', linewidth=1)

plt.axhline(y=len(SAT_D47_columns_monthly), color='black', linestyle='--', linewidth=1)
plt.axhline(y=len(SAT_D47_columns_monthly) + len(SST_D47_columns_monthly), color='black', linestyle='--', linewidth=1)
plt.axhline(y=len(SAT_D47_columns_monthly) + len(SST_D47_columns_monthly) + len(d18Oc_Harwood_columns_monthly), color='black', linestyle='--', linewidth=1)

# Add parameter labels
plt.text(len(SAT_D47_columns_monthly) / 2, -2, 'D47 value from SAT', ha='center', va='center', fontsize=10)
plt.text(len(SAT_D47_columns_monthly) + len(SST_D47_columns_monthly) / 2, -2, 'D47 value from SST', ha='center', va='center', fontsize=10)
plt.text(len(SAT_D47_columns_monthly) + len(SST_D47_columns_monthly) + len(d18Oc_Harwood_columns_monthly) / 2, -2, 'd18Oc (following Harwood et al)', ha='center', va='center', fontsize=10)
plt.text(len(SAT_D47_columns_monthly) + len(SST_D47_columns_monthly) + len(d18Oc_Harwood_columns_monthly) + len(precip_columns_monthly) / 2, -2, 'Precipitation', ha='center', va='center', fontsize=10)

plt.text(-2, len(SAT_D47_columns_monthly) / 2, 'D47 value from SAT', ha='center', va='center', rotation=90, fontsize=10)
plt.text(-2, len(SAT_D47_columns_monthly) + len(SST_D47_columns_monthly) / 2, 'D47 value from SST', ha='center', va='center', rotation=90, fontsize=10)
plt.text(-2, len(SAT_D47_columns_monthly) + len(SST_D47_columns_monthly) + len(d18Oc_Harwood_columns_monthly) / 2, 'd18Oc (following Harwood et al)', ha='center', va='center', rotation=90, fontsize=10)
plt.text(-2, len(SAT_D47_columns_monthly) + len(SST_D47_columns_monthly) + len(d18Oc_Harwood_columns_monthly) + len(precip_columns_monthly) / 2, 'Precipitation', ha='center', va='center', rotation=90, fontsize=10)

plt.title("Raw Combined Covariance Matrix")
plt.show()

# Normalize each submatrix independently for better visualization
def normalize_matrix(matrix):
    min_val = np.min(matrix)
    max_val = np.max(matrix)
    return (matrix - min_val) / (max_val - min_val)

# Extract the relevant columns for SAT, SST D47, d18Oc (following Harwood), and precipitation
SAT_D47_columns_monthly = [f"{month}_SAT_D47" for month in months]
SST_D47_columns_monthly = [f"{month}_SST_D47" for month in months]
d18Oc_Harwood_columns_monthly = [f"{month}_d18Oc_Harwood" for month in months]
precip_columns_monthly = [f"{month}_precip" for month in months]


# Extract the covariance matrices for SAT D47, SST D47, d18Oc_Harwood, and precipitation
cov_SAT_D47_monthly = cov_combined_monthly_Harwood[:len(months), :len(months)]
cov_SST_D47_monthly = cov_combined_monthly_Harwood[len(months):2*len(months), len(months):2*len(months)]
cov_d18Oc_Harwood_monthly = cov_combined_monthly_Harwood[2*len(months):3*len(months), 2*len(months):3*len(months)]
cov_precip_monthly = cov_combined_monthly_Harwood[3*len(months):, 3*len(months):]

# Extract the cross-covariance matrices
cross_cov_SAT_SST_D47_monthly = cov_combined_monthly_Harwood[:len(months), len(months):2*len(months)]
cross_cov_SAT_d18Oc_Harwood_monthly = cov_combined_monthly_Harwood[:len(months), 2*len(months):3*len(months)]
cross_cov_SAT_precip_monthly = cov_combined_monthly_Harwood[:len(months), 3*len(months):]
cross_cov_SST_d18Oc_Harwood_monthly = cov_combined_monthly_Harwood[len(months):2*len(months), 2*len(months):3*len(months)]
cross_cov_SST_precip_monthly = cov_combined_monthly_Harwood[len(months):2*len(months), 3*len(months):]
cross_cov_d18Oc_Harwood_precip_monthly = cov_combined_monthly_Harwood[2*len(months):3*len(months), 3*len(months):]

# Normalize each submatrix
normalized_cov_SAT_D47_monthly = normalize_matrix(cov_SAT_D47_monthly)
normalized_cov_SST_D47_monthly = normalize_matrix(cov_SST_D47_monthly)
normalized_cov_d18Oc_Harwood_monthly = normalize_matrix(cov_d18Oc_Harwood_monthly)
normalized_cov_precip_monthly = normalize_matrix(cov_precip_monthly)

# Normalize each cross-covariance matrix
normalized_cross_cov_SAT_SST_D47_monthly = normalize_matrix(cross_cov_SAT_SST_D47_monthly)
normalized_cross_cov_SAT_d18Oc_Harwood_monthly = normalize_matrix(cross_cov_SAT_d18Oc_Harwood_monthly)
normalized_cross_cov_SAT_precip_monthly = normalize_matrix(cross_cov_SAT_precip_monthly)
normalized_cross_cov_SST_d18Oc_Harwood_monthly = normalize_matrix(cross_cov_SST_d18Oc_Harwood_monthly)
normalized_cross_cov_SST_precip_monthly = normalize_matrix(cross_cov_SST_precip_monthly)
normalized_cross_cov_d18Oc_Harwood_precip_monthly = normalize_matrix(cross_cov_d18Oc_Harwood_precip_monthly)

# Combine the normalized submatrices into a single normalized covariance matrix
normalized_cov_combined_monthly_Harwood = np.block([
    [normalized_cov_SAT_D47_monthly, normalized_cross_cov_SAT_SST_D47_monthly, normalized_cross_cov_SAT_d18Oc_Harwood_monthly, normalized_cross_cov_SAT_precip_monthly],
    [normalized_cross_cov_SAT_SST_D47_monthly.T, normalized_cov_SST_D47_monthly, normalized_cross_cov_SST_d18Oc_Harwood_monthly, normalized_cross_cov_SST_precip_monthly],
    [normalized_cross_cov_SAT_d18Oc_Harwood_monthly.T, normalized_cross_cov_SST_d18Oc_Harwood_monthly.T, normalized_cov_d18Oc_Harwood_monthly, normalized_cross_cov_d18Oc_Harwood_precip_monthly],
    [normalized_cross_cov_SAT_precip_monthly.T, normalized_cross_cov_SST_precip_monthly.T, normalized_cross_cov_d18Oc_Harwood_precip_monthly.T, normalized_cov_precip_monthly]
])

# Plot the heatmap of the normalized combined covariance matrix
plt.figure(figsize=(12, 10))
sns.heatmap(
    normalized_cov_combined_monthly_Harwood,
    annot=False,
    fmt=".2f",
    cmap="coolwarm",
    center=0,
    xticklabels=SAT_D47_columns_monthly + SST_D47_columns_monthly + d18Oc_Harwood_columns_monthly + precip_columns_monthly,
    yticklabels=SAT_D47_columns_monthly + SST_D47_columns_monthly + d18Oc_Harwood_columns_monthly + precip_columns_monthly
)

# Add titles to the axes per parameter
plt.axvline(x=len(SAT_D47_columns_monthly), color='black', linestyle='--', linewidth=1)
plt.axvline(x=len(SAT_D47_columns_monthly) + len(SST_D47_columns_monthly), color='black', linestyle='--', linewidth=1)
plt.axvline(x=len(SAT_D47_columns_monthly) + len(SST_D47_columns_monthly) + len(d18Oc_Harwood_columns_monthly), color='black', linestyle='--', linewidth=1)

plt.axhline(y=len(SAT_D47_columns_monthly), color='black', linestyle='--', linewidth=1)
plt.axhline(y=len(SAT_D47_columns_monthly) + len(SST_D47_columns_monthly), color='black', linestyle='--', linewidth=1)
plt.axhline(y=len(SAT_D47_columns_monthly) + len(SST_D47_columns_monthly) + len(d18Oc_Harwood_columns_monthly), color='black', linestyle='--', linewidth=1)

# Add parameter labels
plt.text(len(SAT_D47_columns_monthly) / 2, -2, 'D47 value from SAT', ha='center', va='center', fontsize=10)
plt.text(len(SAT_D47_columns_monthly) + len(SST_D47_columns_monthly) / 2, -2, 'D47 value from SST', ha='center', va='center', fontsize=10)
plt.text(len(SAT_D47_columns_monthly) + len(SST_D47_columns_monthly) + len(d18Oc_Harwood_columns_monthly) / 2, -2, 'd18Oc_Harwood', ha='center', va='center', fontsize=10)
plt.text(len(SAT_D47_columns_monthly) + len(SST_D47_columns_monthly) + len(d18Oc_Harwood_columns_monthly) + len(precip_columns_monthly) / 2, -2, 'Precipitation', ha='center', va='center', fontsize=10)

plt.text(-7, len(SAT_D47_columns_monthly) / 2, 'D47 value from SAT', ha='center', va='center', rotation=90, fontsize=10)
plt.text(-7, len(SAT_D47_columns_monthly) + len(SST_D47_columns_monthly) / 2, 'D47 value from SST', ha='center', va='center', rotation=90, fontsize=10)
plt.text(-7, len(SAT_D47_columns_monthly) + len(SST_D47_columns_monthly) + len(d18Oc_Harwood_columns_monthly) / 2, 'd18Oc_Harwood', ha='center', va='center', rotation=90, fontsize=10)
plt.text(-7, len(SAT_D47_columns_monthly) + len(SST_D47_columns_monthly) + len(d18Oc_Harwood_columns_monthly) + len(precip_columns_monthly) / 2, 'Precipitation', ha='center', va='center', rotation=90, fontsize=10)

plt.title("Normalized Combined Covariance Matrix")
plt.show()

# Combine the prior means of D47 and SAT into a single state vector
mu_prior_monthly_combined_Harwood = np.concatenate((mu_prior_SST_D47_monthly, mu_prior_SAT_D47_monthly, mu_prior_d18Oc_Harwood_monthly, mu_prior_precip_monthly))
mu_prior_monthly_combined_Graniero = np.concatenate((mu_prior_SST_D47_monthly, mu_prior_SAT_D47_monthly, mu_prior_d18Oc_Graniero_monthly, mu_prior_precip_monthly))
mu_prior_monthly_combined_Railsback_ice = np.concatenate((mu_prior_SST_D47_monthly, mu_prior_SAT_D47_monthly, mu_prior_d18Oc_Railsback_ice_monthly, mu_prior_precip_monthly))
mu_prior_monthly_combined_Railsback_green = np.concatenate((mu_prior_SST_D47_monthly, mu_prior_SAT_D47_monthly, mu_prior_d18Oc_Railsback_green_monthly, mu_prior_precip_monthly))

# Combine the covariance matrices of D47 values of SST and SAT, including the cross-covariance
cov_prior_monthly_combined_Harwood = cov_combined_monthly_Harwood.copy()
cov_prior_monthly_combined_Graniero = cov_combined_monthly_Graniero.copy()
cov_prior_monthly_combined_Railsback_ice = cov_combined_monthly_Railsback_ice.copy()
cov_prior_monthly_combined_Railsback_green = cov_combined_monthly_Railsback_green.copy()

# Load seasonal measurements and format them into a dictionary
# This is precompiled seasonal data per specimen and therefore does not come with a time uncertainty
PWP_data_seasonal = pd.read_csv('PWP case/PWP_proxy_data_seasonal.csv') # Load the data for this assignment into Python and in the Jupyter environment.
PWP_data_seasonal_dict = PWP_data_seasonal.to_dict('records') # Convert to dictionary with column headers as keys

# Add an entry for the time uncertainty (which is always zero in this case, because data is already aggregated seasonally)
for record in PWP_data_seasonal_dict:
    record["Season_err"] = 0

# Load measurements and format them into a dictionary
# These are the actual individual D47 measurements and ShellChron outcomes and thus come with a time uncertainty which can be propagated.
PWP_data = pd.read_csv('PWP case/PWP_proxy_data_months.csv') # Load the data for this assignment into Python and in the Jupyter environment.
PWP_data_dict = PWP_data.to_dict('records') # Convert to dictionary with column headers as keys

# Process the ShellChron_DOY_err field and create a new entry for months
for record in PWP_data_dict:
    # Handle missing values and convert from days to months and seasons
    shell_chron_doy_err = record.get("ShellChron_DOY_err", np.nan)  # Get value, default to NaN if missing
    if pd.isna(shell_chron_doy_err):  # Check if the value is NaN
        record["ShellChron_Month_err"] = 0  # Set the new entry to 0 if missing
        record["ShellChron_Season_err"] = 0  # Set the new entry to 0 if missing
        record["ShellChron_no_err"] = 0  # Add a new entry for no error
    else:
        record["ShellChron_Month_err"] = shell_chron_doy_err / 365 * 12  # Convert days to months
        record["ShellChron_Season_err"] = shell_chron_doy_err / 365 * 4  # Convert days to seasons
        record["ShellChron_no_err"] = 0  # Add a new entry for no error
    
    # Add the month_score2 and season_score field based on the ShellChron_DOY field
    ShellChron_DOY = record.get("ShellChron_DOY_rotated", np.nan)  # Get the ShellChron day of the year value
    if not pd.isna(ShellChron_DOY):  # Check if the ShellChron value is not NaN
        record["month_score2"] = ShellChron_DOY / 365 * 12  # Convert DOY value to month_score
        record["season_score"] = ShellChron_DOY / 365 * 4  # Convert DOY value to season_score

# Function to calculate wrapped normal distribution weights
def wrapped_normal_pdf(x, mean, sd, num_bins):
    # Calculate the normal PDF for each bin
    pdf = stats.norm.pdf(x, loc = mean, scale = sd)
    # Wrap around the bins
    for i in range(1, num_bins):
        pdf += stats.norm.pdf(x + i * num_bins, loc = mean, scale = sd)
        pdf += stats.norm.pdf(x - i * num_bins, loc = mean, scale = sd)
    # Normalize the weights to ensure the sum equals 1
    pdf /= pdf.sum()
    return pdf

# Initialize dictionaries to store aggregated measurements and uncertainties per month
D47_measurements_monthly = {month: [] for month in month_names}
d18Oc_measurements_monthly = {month: [] for month in month_names}
D47_measurement_uncertainties_monthly = {month: [] for month in month_names}
d18Oc_measurement_uncertainties_monthly = {month: [] for month in month_names}

# Aggregate measurements and uncertainties per month
for measurement in PWP_data_dict:
    if not pd.isna(measurement["D47_corrETF_strict"]):
        month_index = int(measurement["month_score2"])
        D47_measurements_monthly[month_names[month_index]].append(measurement["D47_corrETF_strict"])
        D47_measurement_uncertainties_monthly[month_names[month_index]].append(measurement["D47_sd_ext"] ** 2)
    if not pd.isna(measurement["Final_d18O"]):
        month_index = int(measurement["month_score2"])
        d18Oc_measurements_monthly[month_names[month_index]].append(measurement["Final_d18O"])
        d18Oc_measurement_uncertainties_monthly[month_names[month_index]].append(measurement["d18O_sd_ext"] ** 2)

# Calculate weighted mean and propagated uncertainty for each month
D47_aggregated_measurements_monthly = []
d18Oc_aggregated_measurements_monthly = []
D47_aggregated_uncertainties_monthly = []
d18Oc_aggregated_uncertainties_monthly = []

for month in month_names:
    if D47_measurements_monthly[month]:
        weights = 1 / np.array(D47_measurement_uncertainties_monthly[month])
        weighted_mean = np.sum(np.array(D47_measurements_monthly[month]) * weights) / np.sum(weights)
        measurement_variance = np.var(D47_measurements_monthly[month], ddof=1)
        propagated_uncertainty = np.sqrt(1 / np.sum(weights) + measurement_variance)
        D47_aggregated_measurements_monthly.append(weighted_mean)
        D47_aggregated_uncertainties_monthly.append(propagated_uncertainty ** 2)
    if d18Oc_measurements_monthly[month]:
        weights = 1 / np.array(d18Oc_measurement_uncertainties_monthly[month])
        weighted_mean = np.sum(np.array(d18Oc_measurements_monthly[month]) * weights) / np.sum(weights)
        measurement_variance = np.var(d18Oc_measurements_monthly[month], ddof=1)
        propagated_uncertainty = np.sqrt(1 / np.sum(weights) + measurement_variance)
        d18Oc_aggregated_measurements_monthly.append(weighted_mean)
        d18Oc_aggregated_uncertainties_monthly.append(propagated_uncertainty ** 2)

# OPTIONAL: Lower boundary d18Oc variance at 0.01 (equivalent to 0.1 per mil measurement uncertainty)
d18Oc_aggregated_uncertainties_monthly = [max(uncertainty, 0.01) for uncertainty in d18Oc_aggregated_uncertainties_monthly]

# Create the measurement matrix Z
Z_monthly_aggregated = np.array(D47_aggregated_measurements_monthly + d18Oc_aggregated_measurements_monthly).reshape(-1, 1)

# Create the measurement uncertainty matrix R (diagonal matrix)
R_monthly_aggregated = np.diag(D47_aggregated_uncertainties_monthly + d18Oc_aggregated_uncertainties_monthly)  # Diagonal matrix of measurement uncertainties

print("Z_monthly_aggregated:", Z_monthly_aggregated)
print("R_monthly_aggregated:", R_monthly_aggregated)

Z_monthly_aggregated: [[0.6286965 ]
 [0.66196597]
 [0.63593138]
 [0.63587406]
 [0.6385765 ]
 [0.63610532]
 [0.64055775]
 [0.58409635]
 [0.64753068]
 [0.65564679]
 [0.62024339]
 [0.63756647]
 [1.92069861]
 [1.77654022]
 [1.38742522]
 [1.20859978]
 [1.52259534]
 [1.52992939]
 [1.52807461]
 [1.50322068]
 [1.44285092]
 [1.47484531]
 [1.83296327]
 [1.88853586]]
R_monthly_aggregated: [[0.00161332 0.         0.         0.         0.         0.
  0.         0.         0.         0.         0.         0.
  0.         0.         0.         0.         0.         0.
  0.         0.         0.         0.         0.         0.        ]
 [0.         0.00296031 0.         0.         0.         0.
  0.         0.         0.         0.         0.         0.
  0.         0.         0.         0.         0.         0.
  0.         0.         0.         0.         0.         0.        ]
 [0.         0.         0.00213726 0.         0.         0.
  0.         0.         0.         0.         0.         0.
  0.         0.         0.         0.         0.         0.
  0.         0.         0.         0.         0.         0.        ]
 [0.         0.         0.         0.00203359 0.         0.
  0.         0.         0.         0.         0.         0.
  0.         0.         0.         0.         0.         0.
  0.         0.         0.         0.         0.         0.        ]
 [0.         0.         0.         0.         0.00249296 0.
  0.         0.         0.         0.         0.         0.
  0.         0.         0.         0.         0.         0.
  0.         0.         0.         0.         0.         0.        ]
 [0.         0.         0.         0.         0.         0.00283142
  0.         0.         0.         0.         0.         0.
  0.         0.         0.         0.         0.         0.
  0.         0.         0.         0.         0.         0.        ]
 [0.         0.         0.         0.         0.         0.
  0.00145981 0.         0.         0.         0.         0.
  0.         0.         0.         0.         0.         0.
  0.         0.         0.         0.         0.         0.        ]
 [0.         0.         0.         0.         0.         0.
  0.         0.00300617 0.         0.         0.         0.
  0.         0.         0.         0.         0.         0.
  0.         0.         0.         0.         0.         0.        ]
 [0.         0.         0.         0.         0.         0.
  0.         0.         0.0016946  0.         0.         0.
  0.         0.         0.         0.         0.         0.
  0.         0.         0.         0.         0.         0.        ]
 [0.         0.         0.         0.         0.         0.
  0.         0.         0.         0.00338888 0.         0.
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  0.         0.         0.         0.         0.00252047 0.
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  0.83361167 0.         0.         0.         0.         0.
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  0.46247889 0.         0.         0.         0.         0.        ]
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# Create the observation matrix H for monthly aggregated data

# Define the number of seasons or months
num_months = 12

# Number of monthly aggregated measurements
N_measurements_monthly_aggregated = len(Z_monthly_aggregated)
H_monthly_aggregated = np.zeros((N_measurements_monthly_aggregated, len(mu_prior_monthly_combined_Harwood)))

# Fill the observation matrix H with ones at the positions corresponding to the measurements
for i, measurement in enumerate(Z_monthly_aggregated):
    if i < N_measurements_monthly_aggregated // 2:
        time_index = i % num_months  # Ensure the index is within the range of months
        H_monthly_aggregated[i, time_index] = 1
    else:
        time_index = i % num_months
        H_monthly_aggregated[i, time_index + 24] = 1

print(H_monthly_aggregated)

[[1. 0. 0. ... 0. 0. 0.]
 [0. 1. 0. ... 0. 0. 0.]
 [0. 0. 1. ... 0. 0. 0.]
 ...
 [0. 0. 0. ... 0. 0. 0.]
 [0. 0. 0. ... 0. 0. 0.]
 [0. 0. 0. ... 0. 0. 0.]]

def kalman_update_block(
    mu_prior: np.ndarray,
    cov_prior: np.ndarray,
    Z: np.ndarray,
    R: np.ndarray,
    H: np.ndarray,
    debug_print: bool = False
):
    """
    Perform a Kalman update step for a block of observations.

    Parameters:
    mu_prior (np.ndarray): The prior mean vector.
    cov_prior (np.ndarray): The prior covariance matrix.
    Z (np.ndarray): The measurement matrix.
    R (np.ndarray): The measurement noise covariance matrix.
    H (np.ndarray): The observation matrix.
    debug_print (bool): If True, print debug statements.

    Returns:
    mu_posterior (np.ndarray): The posterior mean vector.
    cov_posterior (np.ndarray): The posterior covariance matrix.
    """
    if debug_print:
        # Print shapes of key variables for debugging
        print("Shape of cov_prior:", cov_prior.shape)
        print("Shape of H:", H.shape)
        print("Shape of R:", R.shape)
        print("Shape of mu_prior:", mu_prior.shape)
        print("Shape of Z:", Z.shape)

    # Compute the Kalman gain
    K = cov_prior @ H.T @ np.linalg.inv(H @ cov_prior @ H.T + R)
    if debug_print:
        print("Shape of K:", K.shape)

    # In-between steps for debugging
    Y_hat = H @ mu_prior  # Compute the predicted observation
    if debug_print:
        print("Shape of Y_hat:", Y_hat.shape)
    innovation = Z - Y_hat.reshape(-1, 1)  # Compute the innovation
    if debug_print:
        print("Shape of innovation:", innovation.shape)
    kalman_gain = K @ innovation  # Compute the Kalman gain
    if debug_print:
        print("Shape of kalman_gain:", kalman_gain.shape)

    # Update the posterior mean estimate
    mu_posterior = mu_prior + kalman_gain.flatten()

    # Update the posterior covariance estimate
    cov_posterior = cov_prior - K @ H @ cov_prior

    return mu_posterior, cov_posterior

# UPDATED UPDATING SCRIPT TO ACCOMMODATE MULTIPLE VARIABLES

# Create function to keep track of the likelihood statistics and data

# Suppress FutureWarning
# warnings.simplefilter(action = 'ignore', category = FutureWarning)

def likelihood_statistics_multi(
    weighted_sum,
    effective_weights_total,
    n_update,
    data_library,
    measurement,
    timestamp,
    timestamp_sd,
    Variable_names = ["Variable_name1", "Variable_name2"],
    Variable_names_SDs = ["Variable_name_SD1", "Variable_name_SD2"]
):
    """
    Incrementally updates the likelihood statistics for seasonal data.

    Parameters:
    - weighted_sum: list
        List tracking the mean times the effective weight for each time bin and variable.
    - effective_weights_total: list
        List tracking the sum of effective weights for each time bin and variable.
    - n_update: list
        List tracking the number of datapoints for each time bin and variable.
    - data_library: dict
        Dictionary tracking individual data points and their uncertainties.
    - measurement: dict
        A single measurement containing data on multiple variables.
    - timestamp: str
        Key in the measurement dictionary for the timestamp (0-based index).
    - timestamp_sd: float
        Standard deviation of uncertainty in the timestamp.
    - Variable_name: list of str
        Key in the measurement dictionary for the variables (e.g. d18Oc, D47).
    - Variable_name_SD: list of str
        Key in the measurement dictionary for the standard deviation on the variables (e.g. d18Oc, D47).
    """
    # Check if at least one combination of variable name and its SD is present in the measurement
    found = False
    for var_name, var_sd_name in zip(Variable_names, Variable_names_SDs):
        if var_name in measurement and var_sd_name in measurement:
            found = True
            break

    if timestamp in measurement and found:
        # Extract the time and data values from the measurement
        time = measurement[timestamp]
        time_sd = measurement[timestamp_sd]
        # Loop through all variable/SD pairs
        for var_name, var_sd_name in zip(Variable_names, Variable_names_SDs):
            if var_name in measurement and var_sd_name in measurement:
                data_val = measurement[var_name]
                data_sd = measurement[var_sd_name]
                
                # Check if the data is valid
                if not np.isnan(data_val) and not np.isnan(data_sd):
                    # Calculate the weight (inverse of variance)
                    weight = 1 / (data_sd ** 2)

                    # Determine the number of bins
                    num_bins_seasonal = int(len(weighted_sum) / len(Variable_names))
                    # Ensure num_bins_seasonal is an integer
                    bin_indices = np.arange(num_bins_seasonal, dtype=np.float64)

                    # Calculate the probability density for each bin
                    if time_sd == 0:  # Catch cases where the time uncertainty is zero (or unknown)
                        probabilities = np.zeros(num_bins_seasonal, dtype=np.float64)
                        bin_index = int(time) % num_bins_seasonal  # Ensure the bin index is within range
                        probabilities[bin_index] = 1  # Set the probability to 1 for the correct bin
                    else:
                        probabilities = stats.norm.pdf(bin_indices, loc=time, scale=time_sd)  # For non-zero time uncertainty, use a normal distribution
                        probabilities /= probabilities.sum()  # Normalize to ensure the sum of probabilities is 1

                    for i, prob in enumerate(probabilities):  # Loop over all possible bin numbers in the probability vector
                        bin_index = i % num_bins_seasonal  # Wrap around to the first bin if it overflows

                        # Update the weighted sums and sample count
                        effective_weight = weight * prob
                        var_idx = Variable_names.index(var_name)  # Find the index of the variable
                        idx = int(var_idx * num_bins_seasonal + bin_index)  # Unique index for (variable, bin)
                        if weighted_sum[idx] is None:
                            weighted_sum[idx] = 0
                            effective_weights_total[idx] = 0
                        weighted_sum[idx] = weighted_sum[idx] + data_val * effective_weight
                        effective_weights_total[idx] = effective_weights_total[idx] + effective_weight

                    # Update n_update for the correct variable and bin
                    var_idx = Variable_names.index(var_name) # Find the index of the variable
                    n_update[var_idx * num_bins_seasonal + (int(time) % num_bins_seasonal)] += 1  # update sample number per bin and variable

                    # Track individual data points and their uncertainties
                    key = (var_name, int(time)) # Store individual data points in a dictionary with (variable, time) as key
                    if key not in data_library:
                        data_library[key] = []  # Initialize the list for a new (time, var_name) pair
                    data_library[key].append((time_sd, data_val, data_sd))
    return weighted_sum, effective_weights_total, n_update, data_library # Return the updated values

# Update the seasonal prior with all measurements assigned to the seasons according to the ShellChron_DOY_rotated field
monthly_aggregated_Harwood_data = {} # Keep track of datapoints per season
n_update_monthly_aggregated_Harwood = np.concatenate([mu_prior_SST_D47_monthly * 0, mu_prior_d18Oc_Harwood_monthly_original * 0]) # Vector to store sample size per season for confidence interval plotting
weighted_sum_monthly_aggregated_Harwood = np.concatenate([mu_prior_SST_D47_monthly * 0, mu_prior_d18Oc_Harwood_monthly_original * 0]) # Vector to store mean temperature per season for confidence interval plotting
effective_weights_total_monthly_aggregated_Harwood = np.concatenate([mu_prior_SST_D47_monthly * 0, mu_prior_d18Oc_Harwood_monthly_original * 0]) # Vector to store temperature uncertainty per season for confidence interval plotting
mu_likelihood_monthly_aggregated_Harwood = np.concatenate([mu_prior_SST_D47_monthly * 0, mu_prior_d18Oc_Harwood_monthly_original * 0]) # Vector to store mean temperature per season for confidence interval plotting
std_likelihood_monthly_aggregated_Harwood = np.concatenate([mu_prior_SST_D47_monthly * 0, mu_prior_d18Oc_Harwood_monthly_original * 0]) # Vector to store temperature uncertainty per season for confidence interval plotting
var_names = ["D47_corrETF_strict", "Final_d18O"] # List of variable names which are updated
var_SD_names = ["D47_sd_ext", "d18O_sd_ext"] # List of names of variable uncertainties which are updated

# Update the prior with seasonal data using the Kalman filter in block updating form
mu_post_monthly_aggregated_Harwood, cov_post_monthly_aggregated_Harwood = kalman_update_block(
    mu_prior_monthly_combined_Harwood,
    cov_prior_monthly_combined_Harwood,
    Z_monthly_aggregated,
    R_monthly_aggregated,
    H_monthly_aggregated
)

# Extract the updated mean values from the combined state vector
mu_post_SST_D47_monthly_aggregated_Harwood = mu_post_monthly_aggregated_Harwood[:len(mu_prior_SST_D47_monthly)]
mu_post_SAT_D47_monthly_aggregated_Harwood = mu_post_monthly_aggregated_Harwood[len(mu_prior_SST_D47_monthly):2*len(mu_prior_SST_D47_monthly)]
mu_post_d18Oc_Harwood_monthly_aggregated = mu_post_monthly_aggregated_Harwood[2*len(mu_prior_SST_D47_monthly):3*len(mu_prior_SST_D47_monthly)]
mu_post_precip_monthly_aggregated_Harwood = mu_post_monthly_aggregated_Harwood[3*len(mu_prior_d18Oc_Harwood_monthly):]

# Extract the updated covariance matrices from the combined covariance matrix
cov_post_SST_D47_monthly_aggregated_Harwood = cov_post_monthly_aggregated_Harwood[:len(mu_prior_SST_D47_monthly), :len(mu_prior_SST_D47_monthly)]
cov_post_SAT_D47_monthly_aggregated_Harwood = cov_post_monthly_aggregated_Harwood[len(mu_prior_SST_D47_monthly):2*len(mu_prior_SST_D47_monthly), len(mu_prior_SST_D47_monthly):2*len(mu_prior_SST_D47_monthly)]
cov_post_d18Oc_Harwood_monthly_aggregated = cov_post_monthly_aggregated_Harwood[2*len(mu_prior_SST_D47_monthly):3*len(mu_prior_SST_D47_monthly), 2*len(mu_prior_SST_D47_monthly):3*len(mu_prior_SST_D47_monthly)]
cov_post_precip_monthly_aggregated_Harwood = cov_post_monthly_aggregated_Harwood[3*len(mu_prior_d18Oc_Harwood_monthly):, 3*len(mu_prior_d18Oc_Harwood_monthly):]

for measurement in PWP_data_dict: # Loop over measurements
    # Track and update likelihood statistics
    weighted_sum_monthly_aggregated_Harwood, effective_weights_total_monthly_aggregated_Harwood, n_update_monthly_aggregated_Harwood, monthly_aggregated_Harwood_data = likelihood_statistics_multi(
        weighted_sum_monthly_aggregated_Harwood,
        effective_weights_total_monthly_aggregated_Harwood,
        n_update_monthly_aggregated_Harwood,
        monthly_aggregated_Harwood_data,
        measurement,
        timestamp = "month_score2",
        timestamp_sd = "ShellChron_no_err",
        Variable_names = var_names,
        Variable_names_SDs = var_SD_names
    )

# Normalize the weighted_sum_monthly_aggregated_Harwood to obtain weighted mean
# Calculate inverse square root of the effective_weights_total_monthly_aggregated_Harwood to contain the weighted standard deviation
# Print likelihood statistics
print("Likelihood statistics:")
num_vars = len(var_names)  # number of variables (e.g., D47, d18O)
num_bins_monthly = int(len(weighted_sum_monthly_aggregated_Harwood) / num_vars)

for var_idx, var_name in enumerate(var_names):
    print(f"Results for variable: {var_name}")
    for bin_idx in range(num_bins_monthly):
        idx = var_idx * num_bins_monthly + bin_idx
        if effective_weights_total_monthly_aggregated_Harwood[idx] is not None and effective_weights_total_monthly_aggregated_Harwood[idx] != 0:
            mu_likelihood_monthly_aggregated_Harwood[idx] = weighted_sum_monthly_aggregated_Harwood[idx] / effective_weights_total_monthly_aggregated_Harwood[idx]
            std_likelihood_monthly_aggregated_Harwood[idx] = np.sqrt(1 / effective_weights_total_monthly_aggregated_Harwood[idx])
        print(f"  Bin {bin_idx + 1}:")
        print(f"    Weighted Average: {mu_likelihood_monthly_aggregated_Harwood[idx]}")
        print(f"    Aggregated Uncertainty: {std_likelihood_monthly_aggregated_Harwood[idx]}")
        print(f"    Number of Data Points: {n_update_monthly_aggregated_Harwood[idx]}")
    print()

print("Original Prior Mean SST-D47 Seasonal:\n", mu_prior_SST_D47_monthly_original)
print("Original Prior Standard Deviation SST-D47 Seasonal:\n", np.sqrt(np.diag(cov_prior_SST_D47_monthly_original)))
print("Updated Posterior Mean SST-D47 Seasonal:\n", mu_post_SST_D47_monthly_aggregated_Harwood)
print("Updated Posterior Standard Deviation SST-D47 Seasonal:\n", np.sqrt(np.diag(cov_post_SST_D47_monthly_aggregated_Harwood)))
print("Original Prior Mean SAT-D47 Seasonal:\n", mu_prior_SAT_D47_monthly_original)
print("Original Prior Standard Deviation SAT-D47 Seasonal:\n", np.sqrt(np.diag(cov_prior_SAT_D47_monthly_original)))
print("Updated Posterior Mean SAT-D47 Seasonal:\n", mu_post_SAT_D47_monthly_aggregated_Harwood)
print("Updated Posterior Standard Deviation SAT-D47 Seasonal:\n", np.sqrt(np.diag(cov_post_SAT_D47_monthly_aggregated_Harwood)))
print("Original Prior Mean d18Oc Seasonal:\n", mu_prior_d18Oc_Harwood_monthly_original)
print("Original Prior Standard Deviation d18Oc Seasonal:\n", np.sqrt(np.diag(cov_prior_d18Oc_Harwood_monthly_original)))
print("Updated Posterior Mean d18Oc Seasonal:\n", mu_post_d18Oc_Harwood_monthly_aggregated)
print("Updated Posterior Standard Deviation d18Oc Seasonal:\n", np.sqrt(np.diag(cov_post_d18Oc_Harwood_monthly_aggregated)))
print("Original Prior Mean precipitation Seasonal:\n", mu_prior_precip_monthly_original)
print("Original Prior Standard Deviation precipitation Seasonal:\n", np.sqrt(np.diag(cov_prior_precip_monthly_original)))
print("Updated Posterior Mean precipitation Seasonal:\n", mu_post_precip_monthly_aggregated_Harwood)
print("Updated Posterior Standard Deviation precipitation Seasonal:\n", np.sqrt(np.diag(cov_post_precip_monthly_aggregated_Harwood)))

Likelihood statistics:
Results for variable: D47_corrETF_strict
  Bin 1:
    Weighted Average: 0.6286965046662804
    Aggregated Uncertainty: 0.006172172468534013
    Number of Data Points: 41.0
  Bin 2:
    Weighted Average: 0.6619659675289109
    Aggregated Uncertainty: 0.005702205186276009
    Number of Data Points: 50.0
  Bin 3:
    Weighted Average: 0.6359313809752828
    Aggregated Uncertainty: 0.005962213370555013
    Number of Data Points: 47.0
  Bin 4:
    Weighted Average: 0.635874055245299
    Aggregated Uncertainty: 0.004533472087358248
    Number of Data Points: 77.0
  Bin 5:
    Weighted Average: 0.6385765019088617
    Aggregated Uncertainty: 0.005970373422008295
    Number of Data Points: 46.0
  Bin 6:
    Weighted Average: 0.6361053169591214
    Aggregated Uncertainty: 0.006784491839417968
    Number of Data Points: 40.0
  Bin 7:
    Weighted Average: 0.6405577455241105
    Aggregated Uncertainty: 0.005527388083791424
    Number of Data Points: 51.0
  Bin 8:
    Weighted Average: 0.5840963489118072
    Aggregated Uncertainty: 0.010265395806790367
    Number of Data Points: 17.0
  Bin 9:
    Weighted Average: 0.647530683554632
    Aggregated Uncertainty: 0.008711396016215336
    Number of Data Points: 24.0
  Bin 10:
    Weighted Average: 0.6556467928095093
    Aggregated Uncertainty: 0.005381146841428655
    Number of Data Points: 59.0
  Bin 11:
    Weighted Average: 0.6202433870418984
    Aggregated Uncertainty: 0.006478420662451016
    Number of Data Points: 38.0
  Bin 12:
    Weighted Average: 0.637566469343801
    Aggregated Uncertainty: 0.00527035539970059
    Number of Data Points: 59.0

Results for variable: Final_d18O
  Bin 1:
    Weighted Average: 1.9206986119811305
    Aggregated Uncertainty: 0.015992659402039835
    Number of Data Points: 49.0
  Bin 2:
    Weighted Average: 1.7765402233319623
    Aggregated Uncertainty: 0.013676034680239486
    Number of Data Points: 61.0
  Bin 3:
    Weighted Average: 1.3874252157646356
    Aggregated Uncertainty: 0.013760566776158802
    Number of Data Points: 59.0
  Bin 4:
    Weighted Average: 1.208599783523956
    Aggregated Uncertainty: 0.012944256504872338
    Number of Data Points: 88.0
  Bin 5:
    Weighted Average: 1.5225953417235654
    Aggregated Uncertainty: 0.015049599993639142
    Number of Data Points: 57.0
  Bin 6:
    Weighted Average: 1.5299293863692658
    Aggregated Uncertainty: 0.016143007933378542
    Number of Data Points: 46.0
  Bin 7:
    Weighted Average: 1.528074609885994
    Aggregated Uncertainty: 0.014914311604356982
    Number of Data Points: 62.0
  Bin 8:
    Weighted Average: 1.5032206782535602
    Aggregated Uncertainty: 0.02337806658212779
    Number of Data Points: 24.0
  Bin 9:
    Weighted Average: 1.4428509222730799
    Aggregated Uncertainty: 0.021820959649631697
    Number of Data Points: 26.0
  Bin 10:
    Weighted Average: 1.4748453065363636
    Aggregated Uncertainty: 0.01322758001392675
    Number of Data Points: 66.0
  Bin 11:
    Weighted Average: 1.8329632716915862
    Aggregated Uncertainty: 0.015593569681394154
    Number of Data Points: 46.0
  Bin 12:
    Weighted Average: 1.8885358609985254
    Aggregated Uncertainty: 0.012907904651196074
    Number of Data Points: 64.0

Original Prior Mean SST-D47 Seasonal:
 [0.64595815 0.64972969 0.64957918 0.64352257 0.63174847 0.61687649
 0.60495042 0.60168179 0.60710259 0.61713861 0.62862875 0.63877671]
Original Prior Standard Deviation SST-D47 Seasonal:
 [0.00614572 0.00638619 0.00611021 0.00523351 0.00611762 0.0083541
 0.00986818 0.01053415 0.01065771 0.00977107 0.00805759 0.00665147]
Updated Posterior Mean SST-D47 Seasonal:
 [0.64759562 0.65482923 0.65731776 0.65098115 0.64048845 0.63078081
 0.62139909 0.61738894 0.61531078 0.61971954 0.62869732 0.63523367]
Updated Posterior Standard Deviation SST-D47 Seasonal:
 [0.00305117 0.0032142  0.00327527 0.00313446 0.00327901 0.00441211
 0.00526525 0.00542351 0.00486937 0.00488333 0.00445479 0.00324109]
Original Prior Mean SAT-D47 Seasonal:
 [0.6535531  0.65354873 0.64877444 0.63916405 0.62623172 0.61087967
 0.6014159  0.6026324  0.61343328 0.62749031 0.64061857 0.64921573]
Original Prior Standard Deviation SAT-D47 Seasonal:
 [0.0081692  0.00869841 0.00782808 0.00672115 0.00673335 0.00732744
 0.0086457  0.0091529  0.00900611 0.00891738 0.00842026 0.00808888]
Updated Posterior Mean SAT-D47 Seasonal:
 [0.65219932 0.65255464 0.64950977 0.64284563 0.63300211 0.62309737
 0.61678502 0.62063718 0.62964026 0.63768559 0.64600535 0.64961257]
Updated Posterior Standard Deviation SAT-D47 Seasonal:
 [0.00338645 0.0026309  0.00241719 0.00304679 0.00418856 0.00550891
 0.00607383 0.00647338 0.00666466 0.00612011 0.00556858 0.00461424]
Original Prior Mean d18Oc Seasonal:
 [ 1.24955356  1.48821276  1.44246505  0.99731593  0.16354539 -0.89974455
 -1.79850605 -2.05495028 -1.59818088 -0.78594795  0.05669023  0.76574859]
Original Prior Standard Deviation d18Oc Seasonal:
 [1.81280103 1.77733509 1.77427535 1.7740801  1.73946053 1.7191968
 1.70980795 1.76480456 1.80080386 1.82777258 1.88551146 1.84844074]
Updated Posterior Mean d18Oc Seasonal:
 [2.19606995 2.48466331 2.57713665 2.33558613 1.72415915 1.11699711
 0.52663625 0.49304106 0.76727257 1.04727521 1.48985017 1.81805578]
Updated Posterior Standard Deviation d18Oc Seasonal:
 [0.32649919 0.31012216 0.28570879 0.26596772 0.29935423 0.33394726
 0.35193226 0.37931559 0.36459262 0.31654979 0.33194948 0.33618409]
Original Prior Mean precipitation Seasonal:
 [3.35899606 2.79623688 2.18241293 1.75411782 1.52990136 1.37490782
 1.44029882 2.14277374 2.72446826 3.26583779 3.82762136 3.79819106]
Original Prior Standard Deviation precipitation Seasonal:
 [0.43419051 0.4450087  0.45465416 0.31157176 0.20340219 0.28421396
 0.4935758  0.5372476  0.459027   0.3647094  0.31141638 0.5035945 ]
Updated Posterior Mean precipitation Seasonal:
 [4.41662549 3.95277243 3.32317917 2.09133846 1.69267948 1.58026471
 2.26249578 2.64428556 2.79727505 3.86494518 4.39879589 5.10906187]
Updated Posterior Standard Deviation precipitation Seasonal:
 [0.34633238 0.37520104 0.36213538 0.1600492  0.10788772 0.24937286
 0.51208449 0.58675381 0.45845948 0.33855641 0.20583091 0.42324636]

# --- D47 ---

# Plot the updated seasonal posterior for SST_D47
std_post_SST_D47_monthly_aggregated_Harwood = np.sqrt(np.diag(cov_post_SST_D47_monthly_aggregated_Harwood))
std_prior_SST_D47_monthly_original = np.sqrt(np.diag(cov_prior_SST_D47_monthly_original))
var_start_D47_monthly = var_names.index("D47_corrETF_strict") * num_bins_monthly # Determine the start index for the D47 variable
var_end_D47_monthly = var_start_D47_monthly + num_bins_monthly # Determine the end index for the D47 variable
n_update_monthly_aggregated_Harwood_D47 = n_update_monthly_aggregated_Harwood[var_start_D47_monthly:var_end_D47_monthly]  # Extract the number of updates for D47

plt.figure(figsize=(10, 6))

# PRIOR
plt.plot(months_scale, mu_prior_SST_D47_monthly_original, label='Prior Mean (PlioMIP models)', color='b', marker='o')
plt.fill_between(
    months_scale,
    mu_prior_SST_D47_monthly_original - stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_SST_D47_monthly_original / np.sqrt(n_models_monthly),
    mu_prior_SST_D47_monthly_original + stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_SST_D47_monthly_original / np.sqrt(n_models_monthly),
    color='b',
    alpha=0.2,
    label='95% Confidence Interval'
)

# LIKELIHOOD
# Determine the start and end indices for the selected variable to parse information from the likelihood statistics
plt.plot(months_scale, mu_likelihood_monthly_aggregated_Harwood[var_start_D47_monthly:var_end_D47_monthly], label='Likelihood Mean (clumped data)', color='y', marker='o')
plt.fill_between(
    months_scale,
    mu_likelihood_monthly_aggregated_Harwood[var_start_D47_monthly:var_end_D47_monthly] - stats.t.ppf(1 - 0.025, n_update_monthly_aggregated_Harwood_D47) * std_likelihood_monthly_aggregated_Harwood[var_start_D47_monthly:var_end_D47_monthly] / np.sqrt(n_update_monthly_aggregated_Harwood_D47),
    mu_likelihood_monthly_aggregated_Harwood[var_start_D47_monthly:var_end_D47_monthly] + stats.t.ppf(1 - 0.025, n_update_monthly_aggregated_Harwood_D47) * std_likelihood_monthly_aggregated_Harwood[var_start_D47_monthly:var_end_D47_monthly] / np.sqrt(n_update_monthly_aggregated_Harwood_D47),
    color='y',
    alpha=0.2,
    label='95% Confidence Interval'
)
for measurement in PWP_data_dict:
    plt.plot(int(measurement["month_score2"]) + 1, measurement["D47_corrETF_strict"], color="y", marker="o", alpha=0.2)
    plt.errorbar(int(measurement["month_score2"]) + 1, measurement["D47_corrETF_strict"], yerr=measurement["D47_sd_ext"], color="y", alpha=0.2, capsize=5)
secax = plt.gca().secondary_xaxis('top')
secax.set_xticks(months_scale)
secax.set_xticklabels([f"n = {int(n)}" for n in n_update_monthly_aggregated_Harwood_D47])
secax.tick_params(axis='x', rotation=0)

# POSTERIOR
plt.plot(months_scale, mu_post_SST_D47_monthly_aggregated_Harwood, label='Posterior Mean (PlioMIP models + clumped data)', color='r', marker='o')
plt.fill_between(
    months_scale,
    mu_post_SST_D47_monthly_aggregated_Harwood - stats.t.ppf(1 - 0.025, (n_update_monthly_aggregated_Harwood_D47 + n_models_monthly)) * std_post_SST_D47_monthly_aggregated_Harwood / np.sqrt(n_update_monthly_aggregated_Harwood_D47 + n_models_monthly),
    mu_post_SST_D47_monthly_aggregated_Harwood + stats.t.ppf(1 - 0.025, (n_update_monthly_aggregated_Harwood_D47 + n_models_monthly)) * std_post_SST_D47_monthly_aggregated_Harwood / np.sqrt(n_update_monthly_aggregated_Harwood_D47 + n_models_monthly),
    color='r',
    alpha=0.2,
    label='95% Confidence Interval (Posterior)'
)

plt.xticks(months_scale, month_names, rotation=45, ha="right")
plt.title('Prior and posterior Mean and 95% Confidence Interval for Seasonal SST D47 values\n(Based on individual measurements)')
plt.xlabel('Month')
plt.ylabel('SST D47 value')
plt.legend()
plt.grid(True)
plt.show()

# ---d18Oc---

# Plot the updated seasonal posterior for d18Oc
std_post_d18Oc_monthly_aggregated_Harwood = np.sqrt(np.diag(cov_post_d18Oc_Harwood_monthly_aggregated))
std_prior_d18Oc_Harwood_monthly_original = np.sqrt(np.diag(cov_prior_d18Oc_Harwood_monthly_original))
var_start_d18Oc_monthly = var_names.index("Final_d18O") * num_bins_monthly # Determine the start index for the d18Oc variable
var_end_d18Oc_monthly = var_start_d18Oc_monthly + num_bins_monthly # Determine the end index for the d18Oc variable
n_update_monthly_aggregated_Harwood_d18Oc = n_update_monthly_aggregated_Harwood[var_start_d18Oc_monthly:var_end_d18Oc_monthly]  # Extract the number of updates for d18Oc

plt.figure(figsize=(10, 6))

# PRIOR
plt.plot(months_scale, mu_prior_d18Oc_Harwood_monthly_original, label='Prior Mean (PlioMIP models)', color='b', marker='o')
plt.fill_between(
    months_scale,
    mu_prior_d18Oc_Harwood_monthly_original - stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_d18Oc_Harwood_monthly_original / np.sqrt(n_models_monthly),
    mu_prior_d18Oc_Harwood_monthly_original + stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_d18Oc_Harwood_monthly_original / np.sqrt(n_models_monthly),
    color='b',
    alpha=0.2,
    label='95% Confidence Interval'
)

# LIKELIHOOD
# Determine the start and end indices for the selected variable
plt.plot(months_scale, mu_likelihood_monthly_aggregated_Harwood[var_start_d18Oc_monthly:var_end_d18Oc_monthly], label='Likelihood Mean (clumped data)', color='y', marker='o')
plt.fill_between(
    months_scale,
    mu_likelihood_monthly_aggregated_Harwood[var_start_d18Oc_monthly:var_end_d18Oc_monthly] - stats.t.ppf(1 - 0.025, n_update_monthly_aggregated_Harwood_d18Oc) * std_likelihood_monthly_aggregated_Harwood[var_start_d18Oc_monthly:var_end_d18Oc_monthly] / np.sqrt(n_update_monthly_aggregated_Harwood_d18Oc),
    mu_likelihood_monthly_aggregated_Harwood[var_start_d18Oc_monthly:var_end_d18Oc_monthly] + stats.t.ppf(1 - 0.025, n_update_monthly_aggregated_Harwood_d18Oc) * std_likelihood_monthly_aggregated_Harwood[var_start_d18Oc_monthly:var_end_d18Oc_monthly] / np.sqrt(n_update_monthly_aggregated_Harwood_d18Oc),
    color='y',
    alpha=0.2,
    label='95% Confidence Interval'
)
for measurement in PWP_data_dict:
    plt.plot(int(measurement["month_score2"]) + 1, measurement["Final_d18O"], color="y", marker="o", alpha=0.2)
    plt.errorbar(int(measurement["month_score2"]) + 1, measurement["Final_d18O"], yerr=measurement["d18O_sd_ext"], color="y", alpha=0.2, capsize=5)
secax = plt.gca().secondary_xaxis('top')
secax.set_xticks(months_scale)
secax.set_xticklabels([f"n = {int(n)}" for n in n_update_monthly_aggregated_Harwood_d18Oc])
secax.tick_params(axis='x', rotation=0)

# POSTERIOR
plt.plot(months_scale, mu_post_d18Oc_Harwood_monthly_aggregated, label='Posterior Mean (PlioMIP models + clumped data)', color='r', marker='o')
plt.fill_between(
    months_scale,
    mu_post_d18Oc_Harwood_monthly_aggregated - stats.t.ppf(1 - 0.025, (n_update_monthly_aggregated_Harwood_d18Oc + n_models_monthly)) * std_post_d18Oc_monthly_aggregated_Harwood / np.sqrt(n_update_monthly_aggregated_Harwood_d18Oc + n_models_monthly),
    mu_post_d18Oc_Harwood_monthly_aggregated + stats.t.ppf(1 - 0.025, (n_update_monthly_aggregated_Harwood_d18Oc + n_models_monthly)) * std_post_d18Oc_monthly_aggregated_Harwood / np.sqrt(n_update_monthly_aggregated_Harwood_d18Oc + n_models_monthly),
    color='r',
    alpha=0.2,
    label='95% Confidence Interval (Posterior)'
)

plt.xticks(months_scale, month_names, rotation=45, ha="right")
plt.title('Prior and posterior Mean and 95% Confidence Interval for Seasonal d18Oc values\n(Based on individual measurements)')
plt.xlabel('Month')
plt.ylabel('d18Oc value')
plt.legend()
plt.grid(True)
plt.show()

# Update the seasonal prior with all measurements assigned to the seasons according to the ShellChron_DOY_rotated field
monthly_aggregated_Graniero_data = {} # Keep track of datapoints per season
n_update_monthly_aggregated_Graniero = np.concatenate([mu_prior_SST_D47_monthly * 0, mu_prior_d18Oc_Graniero_monthly_original * 0]) # Vector to store sample size per season for confidence interval plotting
weighted_sum_monthly_aggregated_Graniero = np.concatenate([mu_prior_SST_D47_monthly * 0, mu_prior_d18Oc_Graniero_monthly_original * 0]) # Vector to store mean temperature per season for confidence interval plotting
effective_weights_total_monthly_aggregated_Graniero = np.concatenate([mu_prior_SST_D47_monthly * 0, mu_prior_d18Oc_Graniero_monthly_original * 0]) # Vector to store temperature uncertainty per season for confidence interval plotting
mu_likelihood_monthly_aggregated_Graniero = np.concatenate([mu_prior_SST_D47_monthly * 0, mu_prior_d18Oc_Graniero_monthly_original * 0]) # Vector to store mean temperature per season for confidence interval plotting
std_likelihood_monthly_aggregated_Graniero = np.concatenate([mu_prior_SST_D47_monthly * 0, mu_prior_d18Oc_Graniero_monthly_original * 0]) # Vector to store temperature uncertainty per season for confidence interval plotting
var_names = ["D47_corrETF_strict", "Final_d18O"] # List of variable names which are updated
var_SD_names = ["D47_sd_ext", "d18O_sd_ext"] # List of names of variable uncertainties which are updated

# Update the prior with seasonal data using the Kalman filter in block updating form
mu_post_monthly_aggregated_Graniero, cov_post_monthly_aggregated_Graniero = kalman_update_block(
    mu_prior_monthly_combined_Graniero,
    cov_prior_monthly_combined_Graniero,
    Z_monthly_aggregated,
    R_monthly_aggregated,
    H_monthly_aggregated
)

# Extract the updated mean values from the combined state vector
mu_post_SST_D47_monthly_aggregated_Graniero = mu_post_monthly_aggregated_Graniero[:len(mu_prior_SST_D47_monthly)]
mu_post_SAT_D47_monthly_aggregated_Graniero = mu_post_monthly_aggregated_Graniero[len(mu_prior_SST_D47_monthly):2*len(mu_prior_SST_D47_monthly)]
mu_post_d18Oc_Graniero_monthly_aggregated = mu_post_monthly_aggregated_Graniero[2*len(mu_prior_SST_D47_monthly):3*len(mu_prior_SST_D47_monthly)]
mu_post_precip_monthly_aggregated_Graniero = mu_post_monthly_aggregated_Graniero[3*len(mu_prior_d18Oc_Graniero_monthly):]

# Extract the updated covariance matrices from the combined covariance matrix
cov_post_SST_D47_monthly_aggregated_Graniero = cov_post_monthly_aggregated_Graniero[:len(mu_prior_SST_D47_monthly), :len(mu_prior_SST_D47_monthly)]
cov_post_SAT_D47_monthly_aggregated_Graniero = cov_post_monthly_aggregated_Graniero[len(mu_prior_SST_D47_monthly):2*len(mu_prior_SST_D47_monthly), len(mu_prior_SST_D47_monthly):2*len(mu_prior_SST_D47_monthly)]
cov_post_d18Oc_Graniero_monthly_aggregated = cov_post_monthly_aggregated_Graniero[2*len(mu_prior_SST_D47_monthly):3*len(mu_prior_SST_D47_monthly), 2*len(mu_prior_SST_D47_monthly):3*len(mu_prior_SST_D47_monthly)]
cov_post_precip_monthly_aggregated_Graniero = cov_post_monthly_aggregated_Graniero[3*len(mu_prior_d18Oc_Graniero_monthly):, 3*len(mu_prior_d18Oc_Graniero_monthly):]

for measurement in PWP_data_dict: # Loop over measurements
    # Track and update likelihood statistics
    weighted_sum_monthly_aggregated_Graniero, effective_weights_total_monthly_aggregated_Graniero, n_update_monthly_aggregated_Graniero, monthly_aggregated_Graniero_data = likelihood_statistics_multi(
        weighted_sum_monthly_aggregated_Graniero,
        effective_weights_total_monthly_aggregated_Graniero,
        n_update_monthly_aggregated_Graniero,
        monthly_aggregated_Graniero_data,
        measurement,
        timestamp = "month_score2",
        timestamp_sd = "ShellChron_no_err",
        Variable_names = var_names,
        Variable_names_SDs = var_SD_names
    )

# Normalize the weighted_sum_monthly_aggregated_Graniero to obtain weighted mean
# Calculate inverse square root of the effective_weights_total_monthly_aggregated_Graniero to contain the weighted standard deviation
# Print likelihood statistics
print("Likelihood statistics:")
num_vars = len(var_names)  # number of variables (e.g., D47, d18O)
num_bins_monthly = int(len(weighted_sum_monthly_aggregated_Graniero) / num_vars)

for var_idx, var_name in enumerate(var_names):
    print(f"Results for variable: {var_name}")
    for bin_idx in range(num_bins_monthly):
        idx = var_idx * num_bins_monthly + bin_idx
        if effective_weights_total_monthly_aggregated_Graniero[idx] is not None and effective_weights_total_monthly_aggregated_Graniero[idx] != 0:
            mu_likelihood_monthly_aggregated_Graniero[idx] = weighted_sum_monthly_aggregated_Graniero[idx] / effective_weights_total_monthly_aggregated_Graniero[idx]
            std_likelihood_monthly_aggregated_Graniero[idx] = np.sqrt(1 / effective_weights_total_monthly_aggregated_Graniero[idx])
        print(f"  Bin {bin_idx + 1}:")
        print(f"    Weighted Average: {mu_likelihood_monthly_aggregated_Graniero[idx]}")
        print(f"    Aggregated Uncertainty: {std_likelihood_monthly_aggregated_Graniero[idx]}")
        print(f"    Number of Data Points: {n_update_monthly_aggregated_Graniero[idx]}")
    print()

print("Original Prior Mean SST-D47 Seasonal:\n", mu_prior_SST_D47_monthly_original)
print("Original Prior Standard Deviation SST-D47 Seasonal:\n", np.sqrt(np.diag(cov_prior_SST_D47_monthly_original)))
print("Updated Posterior Mean SST-D47 Seasonal:\n", mu_post_SST_D47_monthly_aggregated_Graniero)
print("Updated Posterior Standard Deviation SST-D47 Seasonal:\n", np.sqrt(np.diag(cov_post_SST_D47_monthly_aggregated_Graniero)))
print("Original Prior Mean SAT-D47 Seasonal:\n", mu_prior_SAT_D47_monthly_original)
print("Original Prior Standard Deviation SAT-D47 Seasonal:\n", np.sqrt(np.diag(cov_prior_SAT_D47_monthly_original)))
print("Updated Posterior Mean SAT-D47 Seasonal:\n", mu_post_SAT_D47_monthly_aggregated_Graniero)
print("Updated Posterior Standard Deviation SAT-D47 Seasonal:\n", np.sqrt(np.diag(cov_post_SAT_D47_monthly_aggregated_Graniero)))
print("Original Prior Mean d18Oc Seasonal:\n", mu_prior_d18Oc_Graniero_monthly_original)
print("Original Prior Standard Deviation d18Oc Seasonal:\n", np.sqrt(np.diag(cov_prior_d18Oc_Graniero_monthly_original)))
print("Updated Posterior Mean d18Oc Seasonal:\n", mu_post_d18Oc_Graniero_monthly_aggregated)
print("Updated Posterior Standard Deviation d18Oc Seasonal:\n", np.sqrt(np.diag(cov_post_d18Oc_Graniero_monthly_aggregated)))
print("Original Prior Mean precipitation Seasonal:\n", mu_prior_precip_monthly_original)
print("Original Prior Standard Deviation precipitation Seasonal:\n", np.sqrt(np.diag(cov_prior_precip_monthly_original)))
print("Updated Posterior Mean precipitation Seasonal:\n", mu_post_precip_monthly_aggregated_Graniero)
print("Updated Posterior Standard Deviation precipitation Seasonal:\n", np.sqrt(np.diag(cov_post_precip_monthly_aggregated_Graniero)))

Likelihood statistics:
Results for variable: D47_corrETF_strict
  Bin 1:
    Weighted Average: 0.6286965046662804
    Aggregated Uncertainty: 0.006172172468534013
    Number of Data Points: 41.0
  Bin 2:
    Weighted Average: 0.6619659675289109
    Aggregated Uncertainty: 0.005702205186276009
    Number of Data Points: 50.0
  Bin 3:
    Weighted Average: 0.6359313809752828
    Aggregated Uncertainty: 0.005962213370555013
    Number of Data Points: 47.0
  Bin 4:
    Weighted Average: 0.635874055245299
    Aggregated Uncertainty: 0.004533472087358248
    Number of Data Points: 77.0
  Bin 5:
    Weighted Average: 0.6385765019088617
    Aggregated Uncertainty: 0.005970373422008295
    Number of Data Points: 46.0
  Bin 6:
    Weighted Average: 0.6361053169591214
    Aggregated Uncertainty: 0.006784491839417968
    Number of Data Points: 40.0
  Bin 7:
    Weighted Average: 0.6405577455241105
    Aggregated Uncertainty: 0.005527388083791424
    Number of Data Points: 51.0
  Bin 8:
    Weighted Average: 0.5840963489118072
    Aggregated Uncertainty: 0.010265395806790367
    Number of Data Points: 17.0
  Bin 9:
    Weighted Average: 0.647530683554632
    Aggregated Uncertainty: 0.008711396016215336
    Number of Data Points: 24.0
  Bin 10:
    Weighted Average: 0.6556467928095093
    Aggregated Uncertainty: 0.005381146841428655
    Number of Data Points: 59.0
  Bin 11:
    Weighted Average: 0.6202433870418984
    Aggregated Uncertainty: 0.006478420662451016
    Number of Data Points: 38.0
  Bin 12:
    Weighted Average: 0.637566469343801
    Aggregated Uncertainty: 0.00527035539970059
    Number of Data Points: 59.0

Results for variable: Final_d18O
  Bin 1:
    Weighted Average: 1.9206986119811305
    Aggregated Uncertainty: 0.015992659402039835
    Number of Data Points: 49.0
  Bin 2:
    Weighted Average: 1.7765402233319623
    Aggregated Uncertainty: 0.013676034680239486
    Number of Data Points: 61.0
  Bin 3:
    Weighted Average: 1.3874252157646356
    Aggregated Uncertainty: 0.013760566776158802
    Number of Data Points: 59.0
  Bin 4:
    Weighted Average: 1.208599783523956
    Aggregated Uncertainty: 0.012944256504872338
    Number of Data Points: 88.0
  Bin 5:
    Weighted Average: 1.5225953417235654
    Aggregated Uncertainty: 0.015049599993639142
    Number of Data Points: 57.0
  Bin 6:
    Weighted Average: 1.5299293863692658
    Aggregated Uncertainty: 0.016143007933378542
    Number of Data Points: 46.0
  Bin 7:
    Weighted Average: 1.528074609885994
    Aggregated Uncertainty: 0.014914311604356982
    Number of Data Points: 62.0
  Bin 8:
    Weighted Average: 1.5032206782535602
    Aggregated Uncertainty: 0.02337806658212779
    Number of Data Points: 24.0
  Bin 9:
    Weighted Average: 1.4428509222730799
    Aggregated Uncertainty: 0.021820959649631697
    Number of Data Points: 26.0
  Bin 10:
    Weighted Average: 1.4748453065363636
    Aggregated Uncertainty: 0.01322758001392675
    Number of Data Points: 66.0
  Bin 11:
    Weighted Average: 1.8329632716915862
    Aggregated Uncertainty: 0.015593569681394154
    Number of Data Points: 46.0
  Bin 12:
    Weighted Average: 1.8885358609985254
    Aggregated Uncertainty: 0.012907904651196074
    Number of Data Points: 64.0

Original Prior Mean SST-D47 Seasonal:
 [0.64595815 0.64972969 0.64957918 0.64352257 0.63174847 0.61687649
 0.60495042 0.60168179 0.60710259 0.61713861 0.62862875 0.63877671]
Original Prior Standard Deviation SST-D47 Seasonal:
 [0.00614572 0.00638619 0.00611021 0.00523351 0.00611762 0.0083541
 0.00986818 0.01053415 0.01065771 0.00977107 0.00805759 0.00665147]
Updated Posterior Mean SST-D47 Seasonal:
 [0.64760239 0.65485809 0.65739096 0.6510295  0.64054718 0.6308409
 0.62143447 0.6173507  0.61526272 0.61966289 0.62863467 0.63519489]
Updated Posterior Standard Deviation SST-D47 Seasonal:
 [0.00307951 0.00323765 0.00326002 0.00313331 0.00328778 0.00442232
 0.00528316 0.00546894 0.0049494  0.00497902 0.00454527 0.00329954]
Original Prior Mean SAT-D47 Seasonal:
 [0.6535531  0.65354873 0.64877444 0.63916405 0.62623172 0.61087967
 0.6014159  0.6026324  0.61343328 0.62749031 0.64061857 0.64921573]
Original Prior Standard Deviation SAT-D47 Seasonal:
 [0.0081692  0.00869841 0.00782808 0.00672115 0.00673335 0.00732744
 0.0086457  0.0091529  0.00900611 0.00891738 0.00842026 0.00808888]
Updated Posterior Mean SAT-D47 Seasonal:
 [0.65220765 0.652521   0.64946695 0.64281597 0.63297781 0.62305828
 0.61671922 0.62054986 0.62959526 0.63772265 0.64609724 0.64967236]
Updated Posterior Standard Deviation SAT-D47 Seasonal:
 [0.00345199 0.0027135  0.00250923 0.00313635 0.00429101 0.0056302
 0.006204   0.00658906 0.00676321 0.00619157 0.00561015 0.00466827]
Original Prior Mean d18Oc Seasonal:
 [ 1.3438645   1.58320839  1.54290413  1.10686585  0.28345529 -0.77220851
 -1.66492102 -1.91919395 -1.46922561 -0.67007851  0.16162142  0.86311472]
Original Prior Standard Deviation d18Oc Seasonal:
 [1.50177344 1.46445261 1.4599008  1.45968833 1.42846825 1.41480843
 1.40968928 1.46695413 1.50185947 1.52472745 1.58016894 1.54079378]
Updated Posterior Mean d18Oc Seasonal:
 [2.18044009 2.46312864 2.55643494 2.32323382 1.72111773 1.11131599
 0.51674663 0.48036855 0.75596807 1.04119498 1.48489893 1.80980212]
Updated Posterior Standard Deviation d18Oc Seasonal:
 [0.32364324 0.30686044 0.28364351 0.26374466 0.29399971 0.33018354
 0.34966934 0.37567726 0.36179724 0.31446659 0.33018412 0.33446271]
Original Prior Mean precipitation Seasonal:
 [3.35899606 2.79623688 2.18241293 1.75411782 1.52990136 1.37490782
 1.44029882 2.14277374 2.72446826 3.26583779 3.82762136 3.79819106]
Original Prior Standard Deviation precipitation Seasonal:
 [0.43419051 0.4450087  0.45465416 0.31157176 0.20340219 0.28421396
 0.4935758  0.5372476  0.459027   0.3647094  0.31141638 0.5035945 ]
Updated Posterior Mean precipitation Seasonal:
 [4.41568361 3.95672924 3.32945972 2.09592277 1.69476382 1.57806201
 2.25096158 2.62897541 2.81255836 3.88455374 4.40402501 5.1074245 ]
Updated Posterior Standard Deviation precipitation Seasonal:
 [0.34837542 0.37531925 0.36032945 0.15891796 0.10851331 0.25110332
 0.516191   0.59170424 0.45689411 0.3335027  0.20510375 0.42467614]

# --- D47 ---

# Plot the updated seasonal posterior for SST_D47
std_post_SST_D47_monthly_aggregated_Graniero = np.sqrt(np.diag(cov_post_SST_D47_monthly_aggregated_Graniero))
std_prior_SST_D47_monthly_original = np.sqrt(np.diag(cov_prior_SST_D47_monthly_original))
var_start_D47_monthly = var_names.index("D47_corrETF_strict") * num_bins_monthly # Determine the start index for the D47 variable
var_end_D47_monthly = var_start_D47_monthly + num_bins_monthly # Determine the end index for the D47 variable
n_update_monthly_aggregated_Graniero_D47 = n_update_monthly_aggregated_Graniero[var_start_D47_monthly:var_end_D47_monthly]  # Extract the number of updates for D47

plt.figure(figsize=(10, 6))

# PRIOR
plt.plot(months_scale, mu_prior_SST_D47_monthly_original, label='Prior Mean (PlioMIP models)', color='b', marker='o')
plt.fill_between(
    months_scale,
    mu_prior_SST_D47_monthly_original - stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_SST_D47_monthly_original / np.sqrt(n_models_monthly),
    mu_prior_SST_D47_monthly_original + stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_SST_D47_monthly_original / np.sqrt(n_models_monthly),
    color='b',
    alpha=0.2,
    label='95% Confidence Interval'
)

# LIKELIHOOD
# Determine the start and end indices for the selected variable to parse information from the likelihood statistics
plt.plot(months_scale, mu_likelihood_monthly_aggregated_Graniero[var_start_D47_monthly:var_end_D47_monthly], label='Likelihood Mean (clumped data)', color='y', marker='o')
plt.fill_between(
    months_scale,
    mu_likelihood_monthly_aggregated_Graniero[var_start_D47_monthly:var_end_D47_monthly] - stats.t.ppf(1 - 0.025, n_update_monthly_aggregated_Graniero_D47) * std_likelihood_monthly_aggregated_Graniero[var_start_D47_monthly:var_end_D47_monthly] / np.sqrt(n_update_monthly_aggregated_Graniero_D47),
    mu_likelihood_monthly_aggregated_Graniero[var_start_D47_monthly:var_end_D47_monthly] + stats.t.ppf(1 - 0.025, n_update_monthly_aggregated_Graniero_D47) * std_likelihood_monthly_aggregated_Graniero[var_start_D47_monthly:var_end_D47_monthly] / np.sqrt(n_update_monthly_aggregated_Graniero_D47),
    color='y',
    alpha=0.2,
    label='95% Confidence Interval'
)
for measurement in PWP_data_dict:
    plt.plot(int(measurement["month_score2"]) + 1, measurement["D47_corrETF_strict"], color="y", marker="o", alpha=0.2)
    plt.errorbar(int(measurement["month_score2"]) + 1, measurement["D47_corrETF_strict"], yerr=measurement["D47_sd_ext"], color="y", alpha=0.2, capsize=5)
secax = plt.gca().secondary_xaxis('top')
secax.set_xticks(months_scale)
secax.set_xticklabels([f"n = {int(n)}" for n in n_update_monthly_aggregated_Graniero_D47])
secax.tick_params(axis='x', rotation=0)

# POSTERIOR
plt.plot(months_scale, mu_post_SST_D47_monthly_aggregated_Graniero, label='Posterior Mean (PlioMIP models + clumped data)', color='r', marker='o')
plt.fill_between(
    months_scale,
    mu_post_SST_D47_monthly_aggregated_Graniero - stats.t.ppf(1 - 0.025, (n_update_monthly_aggregated_Graniero_D47 + n_models_monthly)) * std_post_SST_D47_monthly_aggregated_Graniero / np.sqrt(n_update_monthly_aggregated_Graniero_D47 + n_models_monthly),
    mu_post_SST_D47_monthly_aggregated_Graniero + stats.t.ppf(1 - 0.025, (n_update_monthly_aggregated_Graniero_D47 + n_models_monthly)) * std_post_SST_D47_monthly_aggregated_Graniero / np.sqrt(n_update_monthly_aggregated_Graniero_D47 + n_models_monthly),
    color='r',
    alpha=0.2,
    label='95% Confidence Interval (Posterior)'
)

plt.xticks(months_scale, month_names, rotation=45, ha="right")
plt.title('Prior and posterior Mean and 95% Confidence Interval for Seasonal SST D47 values\n(Based on individual measurements)')
plt.xlabel('Month')
plt.ylabel('SST D47 value')
plt.legend()
plt.grid(True)
plt.show()

# ---d18Oc---

# Plot the updated seasonal posterior for d18Oc
std_post_d18Oc_monthly_aggregated_Graniero = np.sqrt(np.diag(cov_post_d18Oc_Graniero_monthly_aggregated))
std_prior_d18Oc_Graniero_monthly_original = np.sqrt(np.diag(cov_prior_d18Oc_Graniero_monthly_original))
var_start_d18Oc_monthly = var_names.index("Final_d18O") * num_bins_monthly # Determine the start index for the d18Oc variable
var_end_d18Oc_monthly = var_start_d18Oc_monthly + num_bins_monthly # Determine the end index for the d18Oc variable
n_update_monthly_aggregated_Graniero_d18Oc = n_update_monthly_aggregated_Graniero[var_start_d18Oc_monthly:var_end_d18Oc_monthly]  # Extract the number of updates for d18Oc

plt.figure(figsize=(10, 6))

# PRIOR
plt.plot(months_scale, mu_prior_d18Oc_Graniero_monthly_original, label='Prior Mean (PlioMIP models)', color='b', marker='o')
plt.fill_between(
    months_scale,
    mu_prior_d18Oc_Graniero_monthly_original - stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_d18Oc_Graniero_monthly_original / np.sqrt(n_models_monthly),
    mu_prior_d18Oc_Graniero_monthly_original + stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_d18Oc_Graniero_monthly_original / np.sqrt(n_models_monthly),
    color='b',
    alpha=0.2,
    label='95% Confidence Interval'
)

# LIKELIHOOD
# Determine the start and end indices for the selected variable
plt.plot(months_scale, mu_likelihood_monthly_aggregated_Graniero[var_start_d18Oc_monthly:var_end_d18Oc_monthly], label='Likelihood Mean (clumped data)', color='y', marker='o')
plt.fill_between(
    months_scale,
    mu_likelihood_monthly_aggregated_Graniero[var_start_d18Oc_monthly:var_end_d18Oc_monthly] - stats.t.ppf(1 - 0.025, n_update_monthly_aggregated_Graniero_d18Oc) * std_likelihood_monthly_aggregated_Graniero[var_start_d18Oc_monthly:var_end_d18Oc_monthly] / np.sqrt(n_update_monthly_aggregated_Graniero_d18Oc),
    mu_likelihood_monthly_aggregated_Graniero[var_start_d18Oc_monthly:var_end_d18Oc_monthly] + stats.t.ppf(1 - 0.025, n_update_monthly_aggregated_Graniero_d18Oc) * std_likelihood_monthly_aggregated_Graniero[var_start_d18Oc_monthly:var_end_d18Oc_monthly] / np.sqrt(n_update_monthly_aggregated_Graniero_d18Oc),
    color='y',
    alpha=0.2,
    label='95% Confidence Interval'
)
for measurement in PWP_data_dict:
    plt.plot(int(measurement["month_score2"]) + 1, measurement["Final_d18O"], color="y", marker="o", alpha=0.2)
    plt.errorbar(int(measurement["month_score2"]) + 1, measurement["Final_d18O"], yerr=measurement["d18O_sd_ext"], color="y", alpha=0.2, capsize=5)
secax = plt.gca().secondary_xaxis('top')
secax.set_xticks(months_scale)
secax.set_xticklabels([f"n = {int(n)}" for n in n_update_monthly_aggregated_Graniero_d18Oc])
secax.tick_params(axis='x', rotation=0)

# POSTERIOR
plt.plot(months_scale, mu_post_d18Oc_Graniero_monthly_aggregated, label='Posterior Mean (PlioMIP models + clumped data)', color='r', marker='o')
plt.fill_between(
    months_scale,
    mu_post_d18Oc_Graniero_monthly_aggregated - stats.t.ppf(1 - 0.025, (n_update_monthly_aggregated_Graniero_d18Oc + n_models_monthly)) * std_post_d18Oc_monthly_aggregated_Graniero / np.sqrt(n_update_monthly_aggregated_Graniero_d18Oc + n_models_monthly),
    mu_post_d18Oc_Graniero_monthly_aggregated + stats.t.ppf(1 - 0.025, (n_update_monthly_aggregated_Graniero_d18Oc + n_models_monthly)) * std_post_d18Oc_monthly_aggregated_Graniero / np.sqrt(n_update_monthly_aggregated_Graniero_d18Oc + n_models_monthly),
    color='r',
    alpha=0.2,
    label='95% Confidence Interval (Posterior)'
)

plt.xticks(months_scale, month_names, rotation=45, ha="right")
plt.title('Prior and posterior Mean and 95% Confidence Interval for Seasonal d18Oc values\n(Based on individual measurements)')
plt.xlabel('Month')
plt.ylabel('d18Oc value')
plt.legend()
plt.grid(True)
plt.show()

c:\ProgramData\anaconda3\Lib\site-packages\IPython\core\pylabtools.py:170: UserWarning: Creating legend with loc="best" can be slow with large amounts of data.
  fig.canvas.print_figure(bytes_io, **kw)

# Update the seasonal prior with all measurements assigned to the seasons according to the ShellChron_DOY_rotated field
monthly_aggregated_Railsback_ice_data = {} # Keep track of datapoints per season
n_update_monthly_aggregated_Railsback_ice = np.concatenate([mu_prior_SST_D47_monthly * 0, mu_prior_d18Oc_Railsback_ice_monthly_original * 0]) # Vector to store sample size per season for confidence interval plotting
weighted_sum_monthly_aggregated_Railsback_ice = np.concatenate([mu_prior_SST_D47_monthly * 0, mu_prior_d18Oc_Railsback_ice_monthly_original * 0]) # Vector to store mean temperature per season for confidence interval plotting
effective_weights_total_monthly_aggregated_Railsback_ice = np.concatenate([mu_prior_SST_D47_monthly * 0, mu_prior_d18Oc_Railsback_ice_monthly_original * 0]) # Vector to store temperature uncertainty per season for confidence interval plotting
mu_likelihood_monthly_aggregated_Railsback_ice = np.concatenate([mu_prior_SST_D47_monthly * 0, mu_prior_d18Oc_Railsback_ice_monthly_original * 0]) # Vector to store mean temperature per season for confidence interval plotting
std_likelihood_monthly_aggregated_Railsback_ice = np.concatenate([mu_prior_SST_D47_monthly * 0, mu_prior_d18Oc_Railsback_ice_monthly_original * 0]) # Vector to store temperature uncertainty per season for confidence interval plotting
var_names = ["D47_corrETF_strict", "Final_d18O"] # List of variable names which are updated
var_SD_names = ["D47_sd_ext", "d18O_sd_ext"] # List of names of variable uncertainties which are updated

# Update the prior with seasonal data using the Kalman filter in block updating form
mu_post_monthly_aggregated_Railsback_ice, cov_post_monthly_aggregated_Railsback_ice = kalman_update_block(
    mu_prior_monthly_combined_Railsback_ice,
    cov_prior_monthly_combined_Railsback_ice,
    Z_monthly_aggregated,
    R_monthly_aggregated,
    H_monthly_aggregated
)

# Extract the updated mean values from the combined state vector
mu_post_SST_D47_monthly_aggregated_Railsback_ice = mu_post_monthly_aggregated_Railsback_ice[:len(mu_prior_SST_D47_monthly)]
mu_post_SAT_D47_monthly_aggregated_Railsback_ice = mu_post_monthly_aggregated_Railsback_ice[len(mu_prior_SST_D47_monthly):2*len(mu_prior_SST_D47_monthly)]
mu_post_d18Oc_Railsback_ice_monthly_aggregated = mu_post_monthly_aggregated_Railsback_ice[2*len(mu_prior_SST_D47_monthly):3*len(mu_prior_SST_D47_monthly)]
mu_post_precip_monthly_aggregated_Railsback_ice = mu_post_monthly_aggregated_Railsback_ice[3*len(mu_prior_d18Oc_Railsback_ice_monthly):]

# Extract the updated covariance matrices from the combined covariance matrix
cov_post_SST_D47_monthly_aggregated_Railsback_ice = cov_post_monthly_aggregated_Railsback_ice[:len(mu_prior_SST_D47_monthly), :len(mu_prior_SST_D47_monthly)]
cov_post_SAT_D47_monthly_aggregated_Railsback_ice = cov_post_monthly_aggregated_Railsback_ice[len(mu_prior_SST_D47_monthly):2*len(mu_prior_SST_D47_monthly), len(mu_prior_SST_D47_monthly):2*len(mu_prior_SST_D47_monthly)]
cov_post_d18Oc_Railsback_ice_monthly_aggregated = cov_post_monthly_aggregated_Railsback_ice[2*len(mu_prior_SST_D47_monthly):3*len(mu_prior_SST_D47_monthly), 2*len(mu_prior_SST_D47_monthly):3*len(mu_prior_SST_D47_monthly)]
cov_post_precip_monthly_aggregated_Railsback_ice = cov_post_monthly_aggregated_Railsback_ice[3*len(mu_prior_d18Oc_Railsback_ice_monthly):, 3*len(mu_prior_d18Oc_Railsback_ice_monthly):]

for measurement in PWP_data_dict: # Loop over measurements
    # Track and update likelihood statistics
    weighted_sum_monthly_aggregated_Railsback_ice, effective_weights_total_monthly_aggregated_Railsback_ice, n_update_monthly_aggregated_Railsback_ice, monthly_aggregated_Railsback_ice_data = likelihood_statistics_multi(
        weighted_sum_monthly_aggregated_Railsback_ice,
        effective_weights_total_monthly_aggregated_Railsback_ice,
        n_update_monthly_aggregated_Railsback_ice,
        monthly_aggregated_Railsback_ice_data,
        measurement,
        timestamp = "month_score2",
        timestamp_sd = "ShellChron_no_err",
        Variable_names = var_names,
        Variable_names_SDs = var_SD_names
    )

# Normalize the weighted_sum_monthly_aggregated_Railsback_ice to obtain weighted mean
# Calculate inverse square root of the effective_weights_total_monthly_aggregated_Railsback_ice to contain the weighted standard deviation
# Print likelihood statistics
print("Likelihood statistics:")
num_vars = len(var_names)  # number of variables (e.g., D47, d18O)
num_bins_monthly = int(len(weighted_sum_monthly_aggregated_Railsback_ice) / num_vars)

for var_idx, var_name in enumerate(var_names):
    print(f"Results for variable: {var_name}")
    for bin_idx in range(num_bins_monthly):
        idx = var_idx * num_bins_monthly + bin_idx
        if effective_weights_total_monthly_aggregated_Railsback_ice[idx] is not None and effective_weights_total_monthly_aggregated_Railsback_ice[idx] != 0:
            mu_likelihood_monthly_aggregated_Railsback_ice[idx] = weighted_sum_monthly_aggregated_Railsback_ice[idx] / effective_weights_total_monthly_aggregated_Railsback_ice[idx]
            std_likelihood_monthly_aggregated_Railsback_ice[idx] = np.sqrt(1 / effective_weights_total_monthly_aggregated_Railsback_ice[idx])
        print(f"  Bin {bin_idx + 1}:")
        print(f"    Weighted Average: {mu_likelihood_monthly_aggregated_Railsback_ice[idx]}")
        print(f"    Aggregated Uncertainty: {std_likelihood_monthly_aggregated_Railsback_ice[idx]}")
        print(f"    Number of Data Points: {n_update_monthly_aggregated_Railsback_ice[idx]}")
    print()

print("Original Prior Mean SST-D47 Seasonal:\n", mu_prior_SST_D47_monthly_original)
print("Original Prior Standard Deviation SST-D47 Seasonal:\n", np.sqrt(np.diag(cov_prior_SST_D47_monthly_original)))
print("Updated Posterior Mean SST-D47 Seasonal:\n", mu_post_SST_D47_monthly_aggregated_Railsback_ice)
print("Updated Posterior Standard Deviation SST-D47 Seasonal:\n", np.sqrt(np.diag(cov_post_SST_D47_monthly_aggregated_Railsback_ice)))
print("Original Prior Mean SAT-D47 Seasonal:\n", mu_prior_SAT_D47_monthly_original)
print("Original Prior Standard Deviation SAT-D47 Seasonal:\n", np.sqrt(np.diag(cov_prior_SAT_D47_monthly_original)))
print("Updated Posterior Mean SAT-D47 Seasonal:\n", mu_post_SAT_D47_monthly_aggregated_Railsback_ice)
print("Updated Posterior Standard Deviation SAT-D47 Seasonal:\n", np.sqrt(np.diag(cov_post_SAT_D47_monthly_aggregated_Railsback_ice)))
print("Original Prior Mean d18Oc Seasonal:\n", mu_prior_d18Oc_Railsback_ice_monthly_original)
print("Original Prior Standard Deviation d18Oc Seasonal:\n", np.sqrt(np.diag(cov_prior_d18Oc_Railsback_ice_monthly_original)))
print("Updated Posterior Mean d18Oc Seasonal:\n", mu_post_d18Oc_Railsback_ice_monthly_aggregated)
print("Updated Posterior Standard Deviation d18Oc Seasonal:\n", np.sqrt(np.diag(cov_post_d18Oc_Railsback_ice_monthly_aggregated)))
print("Original Prior Mean precipitation Seasonal:\n", mu_prior_precip_monthly_original)
print("Original Prior Standard Deviation precipitation Seasonal:\n", np.sqrt(np.diag(cov_prior_precip_monthly_original)))
print("Updated Posterior Mean precipitation Seasonal:\n", mu_post_precip_monthly_aggregated_Railsback_ice)
print("Updated Posterior Standard Deviation precipitation Seasonal:\n", np.sqrt(np.diag(cov_post_precip_monthly_aggregated_Railsback_ice)))

Likelihood statistics:
Results for variable: D47_corrETF_strict
  Bin 1:
    Weighted Average: 0.6286965046662804
    Aggregated Uncertainty: 0.006172172468534013
    Number of Data Points: 41.0
  Bin 2:
    Weighted Average: 0.6619659675289109
    Aggregated Uncertainty: 0.005702205186276009
    Number of Data Points: 50.0
  Bin 3:
    Weighted Average: 0.6359313809752828
    Aggregated Uncertainty: 0.005962213370555013
    Number of Data Points: 47.0
  Bin 4:
    Weighted Average: 0.635874055245299
    Aggregated Uncertainty: 0.004533472087358248
    Number of Data Points: 77.0
  Bin 5:
    Weighted Average: 0.6385765019088617
    Aggregated Uncertainty: 0.005970373422008295
    Number of Data Points: 46.0
  Bin 6:
    Weighted Average: 0.6361053169591214
    Aggregated Uncertainty: 0.006784491839417968
    Number of Data Points: 40.0
  Bin 7:
    Weighted Average: 0.6405577455241105
    Aggregated Uncertainty: 0.005527388083791424
    Number of Data Points: 51.0
  Bin 8:
    Weighted Average: 0.5840963489118072
    Aggregated Uncertainty: 0.010265395806790367
    Number of Data Points: 17.0
  Bin 9:
    Weighted Average: 0.647530683554632
    Aggregated Uncertainty: 0.008711396016215336
    Number of Data Points: 24.0
  Bin 10:
    Weighted Average: 0.6556467928095093
    Aggregated Uncertainty: 0.005381146841428655
    Number of Data Points: 59.0
  Bin 11:
    Weighted Average: 0.6202433870418984
    Aggregated Uncertainty: 0.006478420662451016
    Number of Data Points: 38.0
  Bin 12:
    Weighted Average: 0.637566469343801
    Aggregated Uncertainty: 0.00527035539970059
    Number of Data Points: 59.0

Results for variable: Final_d18O
  Bin 1:
    Weighted Average: 1.9206986119811305
    Aggregated Uncertainty: 0.015992659402039835
    Number of Data Points: 49.0
  Bin 2:
    Weighted Average: 1.7765402233319623
    Aggregated Uncertainty: 0.013676034680239486
    Number of Data Points: 61.0
  Bin 3:
    Weighted Average: 1.3874252157646356
    Aggregated Uncertainty: 0.013760566776158802
    Number of Data Points: 59.0
  Bin 4:
    Weighted Average: 1.208599783523956
    Aggregated Uncertainty: 0.012944256504872338
    Number of Data Points: 88.0
  Bin 5:
    Weighted Average: 1.5225953417235654
    Aggregated Uncertainty: 0.015049599993639142
    Number of Data Points: 57.0
  Bin 6:
    Weighted Average: 1.5299293863692658
    Aggregated Uncertainty: 0.016143007933378542
    Number of Data Points: 46.0
  Bin 7:
    Weighted Average: 1.528074609885994
    Aggregated Uncertainty: 0.014914311604356982
    Number of Data Points: 62.0
  Bin 8:
    Weighted Average: 1.5032206782535602
    Aggregated Uncertainty: 0.02337806658212779
    Number of Data Points: 24.0
  Bin 9:
    Weighted Average: 1.4428509222730799
    Aggregated Uncertainty: 0.021820959649631697
    Number of Data Points: 26.0
  Bin 10:
    Weighted Average: 1.4748453065363636
    Aggregated Uncertainty: 0.01322758001392675
    Number of Data Points: 66.0
  Bin 11:
    Weighted Average: 1.8329632716915862
    Aggregated Uncertainty: 0.015593569681394154
    Number of Data Points: 46.0
  Bin 12:
    Weighted Average: 1.8885358609985254
    Aggregated Uncertainty: 0.012907904651196074
    Number of Data Points: 64.0

Original Prior Mean SST-D47 Seasonal:
 [0.64595815 0.64972969 0.64957918 0.64352257 0.63174847 0.61687649
 0.60495042 0.60168179 0.60710259 0.61713861 0.62862875 0.63877671]
Original Prior Standard Deviation SST-D47 Seasonal:
 [0.00614572 0.00638619 0.00611021 0.00523351 0.00611762 0.0083541
 0.00986818 0.01053415 0.01065771 0.00977107 0.00805759 0.00665147]
Updated Posterior Mean SST-D47 Seasonal:
 [0.64745066 0.65483419 0.65770842 0.65127981 0.64060021 0.63108606
 0.62180523 0.61798605 0.61538272 0.61940115 0.62827934 0.63467711]
Updated Posterior Standard Deviation SST-D47 Seasonal:
 [0.00294664 0.00310363 0.0032567  0.00307459 0.00318009 0.00427164
 0.00509505 0.00513273 0.00452243 0.0045218  0.00411224 0.00303989]
Original Prior Mean SAT-D47 Seasonal:
 [0.6535531  0.65354873 0.64877444 0.63916405 0.62623172 0.61087967
 0.6014159  0.6026324  0.61343328 0.62749031 0.64061857 0.64921573]
Original Prior Standard Deviation SAT-D47 Seasonal:
 [0.0081692  0.00869841 0.00782808 0.00672115 0.00673335 0.00732744
 0.0086457  0.0091529  0.00900611 0.00891738 0.00842026 0.00808888]
Updated Posterior Mean SAT-D47 Seasonal:
 [0.65147008 0.65184311 0.64890616 0.64236625 0.63245373 0.62262739
 0.61641836 0.62076424 0.6298177  0.63754008 0.64565816 0.64895144]
Updated Posterior Standard Deviation SAT-D47 Seasonal:
 [0.00317735 0.0023948  0.00215639 0.00276842 0.00381645 0.00502417
 0.00556217 0.00598197 0.00623503 0.00582548 0.00538956 0.00442029]
Original Prior Mean d18Oc Seasonal:
 [-0.70238658 -0.46880033 -0.55487901 -1.06753118 -1.97805979 -3.09785252
 -4.04143149 -4.31396305 -3.80680401 -2.89761701 -1.97393634 -1.20882774]
Original Prior Standard Deviation d18Oc Seasonal:
 [4.13280493 4.10382652 4.10968154 4.11474429 4.07333031 4.04541763
 4.03417247 4.08110728 4.11199059 4.13652445 4.19181911 4.15737949]
Updated Posterior Mean d18Oc Seasonal:
 [2.22259543 2.5407927  2.63651195 2.35086938 1.66942227 1.09315823
 0.54033773 0.55627611 0.83888902 1.08119747 1.51526239 1.8192951 ]
Updated Posterior Standard Deviation d18Oc Seasonal:
 [0.34145481 0.32307448 0.29071742 0.27615854 0.32982051 0.35408319
 0.36267054 0.40359042 0.38416385 0.33190809 0.3485269  0.34780865]
Original Prior Mean precipitation Seasonal:
 [3.35899606 2.79623688 2.18241293 1.75411782 1.52990136 1.37490782
 1.44029882 2.14277374 2.72446826 3.26583779 3.82762136 3.79819106]
Original Prior Standard Deviation precipitation Seasonal:
 [0.43419051 0.4450087  0.45465416 0.31157176 0.20340219 0.28421396
 0.4935758  0.5372476  0.459027   0.3647094  0.31141638 0.5035945 ]
Updated Posterior Mean precipitation Seasonal:
 [4.46675097 4.00606715 3.39162767 2.11667603 1.68570577 1.60399986
 2.33948838 2.73936073 2.80621589 3.91281217 4.43771385 5.18817158]
Updated Posterior Standard Deviation precipitation Seasonal:
 [0.32940612 0.36429999 0.35775118 0.16202705 0.1052742  0.23633648
 0.48112069 0.54939646 0.45861164 0.35233741 0.20357078 0.40622614]

# --- D47 ---

# Plot the updated seasonal posterior for SST_D47
std_post_SST_D47_monthly_aggregated_Railsback_ice = np.sqrt(np.diag(cov_post_SST_D47_monthly_aggregated_Railsback_ice))
std_prior_SST_D47_monthly_original = np.sqrt(np.diag(cov_prior_SST_D47_monthly_original))
var_start_D47_monthly = var_names.index("D47_corrETF_strict") * num_bins_monthly # Determine the start index for the D47 variable
var_end_D47_monthly = var_start_D47_monthly + num_bins_monthly # Determine the end index for the D47 variable
n_update_monthly_aggregated_Railsback_ice_D47 = n_update_monthly_aggregated_Railsback_ice[var_start_D47_monthly:var_end_D47_monthly]  # Extract the number of updates for D47

plt.figure(figsize=(10, 6))

# PRIOR
plt.plot(months_scale, mu_prior_SST_D47_monthly_original, label='Prior Mean (PlioMIP models)', color='b', marker='o')
plt.fill_between(
    months_scale,
    mu_prior_SST_D47_monthly_original - stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_SST_D47_monthly_original / np.sqrt(n_models_monthly),
    mu_prior_SST_D47_monthly_original + stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_SST_D47_monthly_original / np.sqrt(n_models_monthly),
    color='b',
    alpha=0.2,
    label='95% Confidence Interval'
)

# LIKELIHOOD
# Determine the start and end indices for the selected variable to parse information from the likelihood statistics
plt.plot(months_scale, mu_likelihood_monthly_aggregated_Railsback_ice[var_start_D47_monthly:var_end_D47_monthly], label='Likelihood Mean (clumped data)', color='y', marker='o')
plt.fill_between(
    months_scale,
    mu_likelihood_monthly_aggregated_Railsback_ice[var_start_D47_monthly:var_end_D47_monthly] - stats.t.ppf(1 - 0.025, n_update_monthly_aggregated_Railsback_ice_D47) * std_likelihood_monthly_aggregated_Railsback_ice[var_start_D47_monthly:var_end_D47_monthly] / np.sqrt(n_update_monthly_aggregated_Railsback_ice_D47),
    mu_likelihood_monthly_aggregated_Railsback_ice[var_start_D47_monthly:var_end_D47_monthly] + stats.t.ppf(1 - 0.025, n_update_monthly_aggregated_Railsback_ice_D47) * std_likelihood_monthly_aggregated_Railsback_ice[var_start_D47_monthly:var_end_D47_monthly] / np.sqrt(n_update_monthly_aggregated_Railsback_ice_D47),
    color='y',
    alpha=0.2,
    label='95% Confidence Interval'
)
for measurement in PWP_data_dict:
    plt.plot(int(measurement["month_score2"]) + 1, measurement["D47_corrETF_strict"], color="y", marker="o", alpha=0.2)
    plt.errorbar(int(measurement["month_score2"]) + 1, measurement["D47_corrETF_strict"], yerr=measurement["D47_sd_ext"], color="y", alpha=0.2, capsize=5)
secax = plt.gca().secondary_xaxis('top')
secax.set_xticks(months_scale)
secax.set_xticklabels([f"n = {int(n)}" for n in n_update_monthly_aggregated_Railsback_ice_D47])
secax.tick_params(axis='x', rotation=0)

# POSTERIOR
plt.plot(months_scale, mu_post_SST_D47_monthly_aggregated_Railsback_ice, label='Posterior Mean (PlioMIP models + clumped data)', color='r', marker='o')
plt.fill_between(
    months_scale,
    mu_post_SST_D47_monthly_aggregated_Railsback_ice - stats.t.ppf(1 - 0.025, (n_update_monthly_aggregated_Railsback_ice_D47 + n_models_monthly)) * std_post_SST_D47_monthly_aggregated_Railsback_ice / np.sqrt(n_update_monthly_aggregated_Railsback_ice_D47 + n_models_monthly),
    mu_post_SST_D47_monthly_aggregated_Railsback_ice + stats.t.ppf(1 - 0.025, (n_update_monthly_aggregated_Railsback_ice_D47 + n_models_monthly)) * std_post_SST_D47_monthly_aggregated_Railsback_ice / np.sqrt(n_update_monthly_aggregated_Railsback_ice_D47 + n_models_monthly),
    color='r',
    alpha=0.2,
    label='95% Confidence Interval (Posterior)'
)

plt.xticks(months_scale, month_names, rotation=45, ha="right")
plt.title('Prior and posterior Mean and 95% Confidence Interval for Seasonal SST D47 values\n(Based on individual measurements)')
plt.xlabel('Month')
plt.ylabel('SST D47 value')
plt.legend()
plt.grid(True)
plt.show()

# ---d18Oc---

# Plot the updated seasonal posterior for d18Oc
std_post_d18Oc_monthly_aggregated_Railsback_ice = np.sqrt(np.diag(cov_post_d18Oc_Railsback_ice_monthly_aggregated))
std_prior_d18Oc_Railsback_ice_monthly_original = np.sqrt(np.diag(cov_prior_d18Oc_Railsback_ice_monthly_original))
var_start_d18Oc_monthly = var_names.index("Final_d18O") * num_bins_monthly # Determine the start index for the d18Oc variable
var_end_d18Oc_monthly = var_start_d18Oc_monthly + num_bins_monthly # Determine the end index for the d18Oc variable
n_update_monthly_aggregated_Railsback_ice_d18Oc = n_update_monthly_aggregated_Railsback_ice[var_start_d18Oc_monthly:var_end_d18Oc_monthly]  # Extract the number of updates for d18Oc

plt.figure(figsize=(10, 6))

# PRIOR
plt.plot(months_scale, mu_prior_d18Oc_Railsback_ice_monthly_original, label='Prior Mean (PlioMIP models)', color='b', marker='o')
plt.fill_between(
    months_scale,
    mu_prior_d18Oc_Railsback_ice_monthly_original - stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_d18Oc_Railsback_ice_monthly_original / np.sqrt(n_models_monthly),
    mu_prior_d18Oc_Railsback_ice_monthly_original + stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_d18Oc_Railsback_ice_monthly_original / np.sqrt(n_models_monthly),
    color='b',
    alpha=0.2,
    label='95% Confidence Interval'
)

# LIKELIHOOD
# Determine the start and end indices for the selected variable
plt.plot(months_scale, mu_likelihood_monthly_aggregated_Railsback_ice[var_start_d18Oc_monthly:var_end_d18Oc_monthly], label='Likelihood Mean (clumped data)', color='y', marker='o')
plt.fill_between(
    months_scale,
    mu_likelihood_monthly_aggregated_Railsback_ice[var_start_d18Oc_monthly:var_end_d18Oc_monthly] - stats.t.ppf(1 - 0.025, n_update_monthly_aggregated_Railsback_ice_d18Oc) * std_likelihood_monthly_aggregated_Railsback_ice[var_start_d18Oc_monthly:var_end_d18Oc_monthly] / np.sqrt(n_update_monthly_aggregated_Railsback_ice_d18Oc),
    mu_likelihood_monthly_aggregated_Railsback_ice[var_start_d18Oc_monthly:var_end_d18Oc_monthly] + stats.t.ppf(1 - 0.025, n_update_monthly_aggregated_Railsback_ice_d18Oc) * std_likelihood_monthly_aggregated_Railsback_ice[var_start_d18Oc_monthly:var_end_d18Oc_monthly] / np.sqrt(n_update_monthly_aggregated_Railsback_ice_d18Oc),
    color='y',
    alpha=0.2,
    label='95% Confidence Interval'
)
for measurement in PWP_data_dict:
    plt.plot(int(measurement["month_score2"]) + 1, measurement["Final_d18O"], color="y", marker="o", alpha=0.2)
    plt.errorbar(int(measurement["month_score2"]) + 1, measurement["Final_d18O"], yerr=measurement["d18O_sd_ext"], color="y", alpha=0.2, capsize=5)
secax = plt.gca().secondary_xaxis('top')
secax.set_xticks(months_scale)
secax.set_xticklabels([f"n = {int(n)}" for n in n_update_monthly_aggregated_Railsback_ice_d18Oc])
secax.tick_params(axis='x', rotation=0)

# POSTERIOR
plt.plot(months_scale, mu_post_d18Oc_Railsback_ice_monthly_aggregated, label='Posterior Mean (PlioMIP models + clumped data)', color='r', marker='o')
plt.fill_between(
    months_scale,
    mu_post_d18Oc_Railsback_ice_monthly_aggregated - stats.t.ppf(1 - 0.025, (n_update_monthly_aggregated_Railsback_ice_d18Oc + n_models_monthly)) * std_post_d18Oc_monthly_aggregated_Railsback_ice / np.sqrt(n_update_monthly_aggregated_Railsback_ice_d18Oc + n_models_monthly),
    mu_post_d18Oc_Railsback_ice_monthly_aggregated + stats.t.ppf(1 - 0.025, (n_update_monthly_aggregated_Railsback_ice_d18Oc + n_models_monthly)) * std_post_d18Oc_monthly_aggregated_Railsback_ice / np.sqrt(n_update_monthly_aggregated_Railsback_ice_d18Oc + n_models_monthly),
    color='r',
    alpha=0.2,
    label='95% Confidence Interval (Posterior)'
)

plt.xticks(months_scale, month_names, rotation=45, ha="right")
plt.title('Prior and posterior Mean and 95% Confidence Interval for Seasonal d18Oc values\n(Based on individual measurements)')
plt.xlabel('Month')
plt.ylabel('d18Oc value')
plt.legend()
plt.grid(True)
plt.show()

# Update the seasonal prior with all measurements assigned to the seasons according to the ShellChron_DOY_rotated field
monthly_aggregated_Railsback_green_data = {} # Keep track of datapoints per season
n_update_monthly_aggregated_Railsback_green = np.concatenate([mu_prior_SST_D47_monthly * 0, mu_prior_d18Oc_Railsback_green_monthly_original * 0]) # Vector to store sample size per season for confidence interval plotting
weighted_sum_monthly_aggregated_Railsback_green = np.concatenate([mu_prior_SST_D47_monthly * 0, mu_prior_d18Oc_Railsback_green_monthly_original * 0]) # Vector to store mean temperature per season for confidence interval plotting
effective_weights_total_monthly_aggregated_Railsback_green = np.concatenate([mu_prior_SST_D47_monthly * 0, mu_prior_d18Oc_Railsback_green_monthly_original * 0]) # Vector to store temperature uncertainty per season for confidence interval plotting
mu_likelihood_monthly_aggregated_Railsback_green = np.concatenate([mu_prior_SST_D47_monthly * 0, mu_prior_d18Oc_Railsback_green_monthly_original * 0]) # Vector to store mean temperature per season for confidence interval plotting
std_likelihood_monthly_aggregated_Railsback_green = np.concatenate([mu_prior_SST_D47_monthly * 0, mu_prior_d18Oc_Railsback_green_monthly_original * 0]) # Vector to store temperature uncertainty per season for confidence interval plotting
var_names = ["D47_corrETF_strict", "Final_d18O"] # List of variable names which are updated
var_SD_names = ["D47_sd_ext", "d18O_sd_ext"] # List of names of variable uncertainties which are updated

# Update the prior with seasonal data using the Kalman filter in block updating form
mu_post_monthly_aggregated_Railsback_green, cov_post_monthly_aggregated_Railsback_green = kalman_update_block(
    mu_prior_monthly_combined_Railsback_green,
    cov_prior_monthly_combined_Railsback_green,
    Z_monthly_aggregated,
    R_monthly_aggregated,
    H_monthly_aggregated
)

# Extract the updated mean values from the combined state vector
mu_post_SST_D47_monthly_aggregated_Railsback_green = mu_post_monthly_aggregated_Railsback_green[:len(mu_prior_SST_D47_monthly)]
mu_post_SAT_D47_monthly_aggregated_Railsback_green = mu_post_monthly_aggregated_Railsback_green[len(mu_prior_SST_D47_monthly):2*len(mu_prior_SST_D47_monthly)]
mu_post_d18Oc_Railsback_green_monthly_aggregated = mu_post_monthly_aggregated_Railsback_green[2*len(mu_prior_SST_D47_monthly):3*len(mu_prior_SST_D47_monthly)]
mu_post_precip_monthly_aggregated_Railsback_green = mu_post_monthly_aggregated_Railsback_green[3*len(mu_prior_d18Oc_Railsback_green_monthly):]

# Extract the updated covariance matrices from the combined covariance matrix
cov_post_SST_D47_monthly_aggregated_Railsback_green = cov_post_monthly_aggregated_Railsback_green[:len(mu_prior_SST_D47_monthly), :len(mu_prior_SST_D47_monthly)]
cov_post_SAT_D47_monthly_aggregated_Railsback_green = cov_post_monthly_aggregated_Railsback_green[len(mu_prior_SST_D47_monthly):2*len(mu_prior_SST_D47_monthly), len(mu_prior_SST_D47_monthly):2*len(mu_prior_SST_D47_monthly)]
cov_post_d18Oc_Railsback_green_monthly_aggregated = cov_post_monthly_aggregated_Railsback_green[2*len(mu_prior_SST_D47_monthly):3*len(mu_prior_SST_D47_monthly), 2*len(mu_prior_SST_D47_monthly):3*len(mu_prior_SST_D47_monthly)]
cov_post_precip_monthly_aggregated_Railsback_green = cov_post_monthly_aggregated_Railsback_green[3*len(mu_prior_d18Oc_Railsback_green_monthly):, 3*len(mu_prior_d18Oc_Railsback_green_monthly):]

for measurement in PWP_data_dict: # Loop over measurements
    # Track and update likelihood statistics
    weighted_sum_monthly_aggregated_Railsback_green, effective_weights_total_monthly_aggregated_Railsback_green, n_update_monthly_aggregated_Railsback_green, monthly_aggregated_Railsback_green_data = likelihood_statistics_multi(
        weighted_sum_monthly_aggregated_Railsback_green,
        effective_weights_total_monthly_aggregated_Railsback_green,
        n_update_monthly_aggregated_Railsback_green,
        monthly_aggregated_Railsback_green_data,
        measurement,
        timestamp = "month_score2",
        timestamp_sd = "ShellChron_no_err",
        Variable_names = var_names,
        Variable_names_SDs = var_SD_names
    )

# Normalize the weighted_sum_monthly_aggregated_Railsback_green to obtain weighted mean
# Calculate inverse square root of the effective_weights_total_monthly_aggregated_Railsback_green to contain the weighted standard deviation
# Print likelihood statistics
print("Likelihood statistics:")
num_vars = len(var_names)  # number of variables (e.g., D47, d18O)
num_bins_monthly = int(len(weighted_sum_monthly_aggregated_Railsback_green) / num_vars)

for var_idx, var_name in enumerate(var_names):
    print(f"Results for variable: {var_name}")
    for bin_idx in range(num_bins_monthly):
        idx = var_idx * num_bins_monthly + bin_idx
        if effective_weights_total_monthly_aggregated_Railsback_green[idx] is not None and effective_weights_total_monthly_aggregated_Railsback_green[idx] != 0:
            mu_likelihood_monthly_aggregated_Railsback_green[idx] = weighted_sum_monthly_aggregated_Railsback_green[idx] / effective_weights_total_monthly_aggregated_Railsback_green[idx]
            std_likelihood_monthly_aggregated_Railsback_green[idx] = np.sqrt(1 / effective_weights_total_monthly_aggregated_Railsback_green[idx])
        print(f"  Bin {bin_idx + 1}:")
        print(f"    Weighted Average: {mu_likelihood_monthly_aggregated_Railsback_green[idx]}")
        print(f"    Aggregated Uncertainty: {std_likelihood_monthly_aggregated_Railsback_green[idx]}")
        print(f"    Number of Data Points: {n_update_monthly_aggregated_Railsback_green[idx]}")
    print()

print("Original Prior Mean SST-D47 Seasonal:\n", mu_prior_SST_D47_monthly_original)
print("Original Prior Standard Deviation SST-D47 Seasonal:\n", np.sqrt(np.diag(cov_prior_SST_D47_monthly_original)))
print("Updated Posterior Mean SST-D47 Seasonal:\n", mu_post_SST_D47_monthly_aggregated_Railsback_green)
print("Updated Posterior Standard Deviation SST-D47 Seasonal:\n", np.sqrt(np.diag(cov_post_SST_D47_monthly_aggregated_Railsback_green)))
print("Original Prior Mean SAT-D47 Seasonal:\n", mu_prior_SAT_D47_monthly_original)
print("Original Prior Standard Deviation SAT-D47 Seasonal:\n", np.sqrt(np.diag(cov_prior_SAT_D47_monthly_original)))
print("Updated Posterior Mean SAT-D47 Seasonal:\n", mu_post_SAT_D47_monthly_aggregated_Railsback_green)
print("Updated Posterior Standard Deviation SAT-D47 Seasonal:\n", np.sqrt(np.diag(cov_post_SAT_D47_monthly_aggregated_Railsback_green)))
print("Original Prior Mean d18Oc Seasonal:\n", mu_prior_d18Oc_Railsback_green_monthly_original)
print("Original Prior Standard Deviation d18Oc Seasonal:\n", np.sqrt(np.diag(cov_prior_d18Oc_Railsback_green_monthly_original)))
print("Updated Posterior Mean d18Oc Seasonal:\n", mu_post_d18Oc_Railsback_green_monthly_aggregated)
print("Updated Posterior Standard Deviation d18Oc Seasonal:\n", np.sqrt(np.diag(cov_post_d18Oc_Railsback_green_monthly_aggregated)))
print("Original Prior Mean precipitation Seasonal:\n", mu_prior_precip_monthly_original)
print("Original Prior Standard Deviation precipitation Seasonal:\n", np.sqrt(np.diag(cov_prior_precip_monthly_original)))
print("Updated Posterior Mean precipitation Seasonal:\n", mu_post_precip_monthly_aggregated_Railsback_green)
print("Updated Posterior Standard Deviation precipitation Seasonal:\n", np.sqrt(np.diag(cov_post_precip_monthly_aggregated_Railsback_green)))

Likelihood statistics:
Results for variable: D47_corrETF_strict
  Bin 1:
    Weighted Average: 0.6286965046662804
    Aggregated Uncertainty: 0.006172172468534013
    Number of Data Points: 41.0
  Bin 2:
    Weighted Average: 0.6619659675289109
    Aggregated Uncertainty: 0.005702205186276009
    Number of Data Points: 50.0
  Bin 3:
    Weighted Average: 0.6359313809752828
    Aggregated Uncertainty: 0.005962213370555013
    Number of Data Points: 47.0
  Bin 4:
    Weighted Average: 0.635874055245299
    Aggregated Uncertainty: 0.004533472087358248
    Number of Data Points: 77.0
  Bin 5:
    Weighted Average: 0.6385765019088617
    Aggregated Uncertainty: 0.005970373422008295
    Number of Data Points: 46.0
  Bin 6:
    Weighted Average: 0.6361053169591214
    Aggregated Uncertainty: 0.006784491839417968
    Number of Data Points: 40.0
  Bin 7:
    Weighted Average: 0.6405577455241105
    Aggregated Uncertainty: 0.005527388083791424
    Number of Data Points: 51.0
  Bin 8:
    Weighted Average: 0.5840963489118072
    Aggregated Uncertainty: 0.010265395806790367
    Number of Data Points: 17.0
  Bin 9:
    Weighted Average: 0.647530683554632
    Aggregated Uncertainty: 0.008711396016215336
    Number of Data Points: 24.0
  Bin 10:
    Weighted Average: 0.6556467928095093
    Aggregated Uncertainty: 0.005381146841428655
    Number of Data Points: 59.0
  Bin 11:
    Weighted Average: 0.6202433870418984
    Aggregated Uncertainty: 0.006478420662451016
    Number of Data Points: 38.0
  Bin 12:
    Weighted Average: 0.637566469343801
    Aggregated Uncertainty: 0.00527035539970059
    Number of Data Points: 59.0

Results for variable: Final_d18O
  Bin 1:
    Weighted Average: 1.9206986119811305
    Aggregated Uncertainty: 0.015992659402039835
    Number of Data Points: 49.0
  Bin 2:
    Weighted Average: 1.7765402233319623
    Aggregated Uncertainty: 0.013676034680239486
    Number of Data Points: 61.0
  Bin 3:
    Weighted Average: 1.3874252157646356
    Aggregated Uncertainty: 0.013760566776158802
    Number of Data Points: 59.0
  Bin 4:
    Weighted Average: 1.208599783523956
    Aggregated Uncertainty: 0.012944256504872338
    Number of Data Points: 88.0
  Bin 5:
    Weighted Average: 1.5225953417235654
    Aggregated Uncertainty: 0.015049599993639142
    Number of Data Points: 57.0
  Bin 6:
    Weighted Average: 1.5299293863692658
    Aggregated Uncertainty: 0.016143007933378542
    Number of Data Points: 46.0
  Bin 7:
    Weighted Average: 1.528074609885994
    Aggregated Uncertainty: 0.014914311604356982
    Number of Data Points: 62.0
  Bin 8:
    Weighted Average: 1.5032206782535602
    Aggregated Uncertainty: 0.02337806658212779
    Number of Data Points: 24.0
  Bin 9:
    Weighted Average: 1.4428509222730799
    Aggregated Uncertainty: 0.021820959649631697
    Number of Data Points: 26.0
  Bin 10:
    Weighted Average: 1.4748453065363636
    Aggregated Uncertainty: 0.01322758001392675
    Number of Data Points: 66.0
  Bin 11:
    Weighted Average: 1.8329632716915862
    Aggregated Uncertainty: 0.015593569681394154
    Number of Data Points: 46.0
  Bin 12:
    Weighted Average: 1.8885358609985254
    Aggregated Uncertainty: 0.012907904651196074
    Number of Data Points: 64.0

Original Prior Mean SST-D47 Seasonal:
 [0.64595815 0.64972969 0.64957918 0.64352257 0.63174847 0.61687649
 0.60495042 0.60168179 0.60710259 0.61713861 0.62862875 0.63877671]
Original Prior Standard Deviation SST-D47 Seasonal:
 [0.00614572 0.00638619 0.00611021 0.00523351 0.00611762 0.0083541
 0.00986818 0.01053415 0.01065771 0.00977107 0.00805759 0.00665147]
Updated Posterior Mean SST-D47 Seasonal:
 [0.64677632 0.65440577 0.65820502 0.65140904 0.64067377 0.63120108
 0.62161741 0.61690041 0.61366877 0.61733156 0.62624652 0.63318016]
Updated Posterior Standard Deviation SST-D47 Seasonal:
 [0.00308057 0.00323847 0.0032593  0.00313315 0.00328797 0.00442251
 0.00528364 0.00547043 0.0049523  0.0049826  0.00454868 0.00330179]
Original Prior Mean SAT-D47 Seasonal:
 [0.6535531  0.65354873 0.64877444 0.63916405 0.62623172 0.61087967
 0.6014159  0.6026324  0.61343328 0.62749031 0.64061857 0.64921573]
Original Prior Standard Deviation SAT-D47 Seasonal:
 [0.0081692  0.00869841 0.00782808 0.00672115 0.00673335 0.00732744
 0.0086457  0.0091529  0.00900611 0.00891738 0.00842026 0.00808888]
Updated Posterior Mean SAT-D47 Seasonal:
 [0.65029711 0.6502254  0.64721022 0.64072545 0.63060024 0.62038804
 0.61382411 0.61819647 0.62770659 0.63629217 0.64524698 0.64823487]
Updated Posterior Standard Deviation SAT-D47 Seasonal:
 [0.00345452 0.00271674 0.00251283 0.00313979 0.00429487 0.00563469
 0.00620883 0.00659328 0.00676676 0.00619415 0.00561162 0.00467029]
Original Prior Mean d18Oc Seasonal:
 [ 0.11114732  0.35051345  0.31038591 -0.12535656 -0.94843075 -2.00384695
 -2.89636307 -3.1505655  -2.70081797 -1.90209574 -1.07075095 -0.36950327]
Original Prior Standard Deviation d18Oc Seasonal:
 [1.49169831 1.4543067  1.44970325 1.44949805 1.41841682 1.40503469
 1.40009942 1.45744644 1.49229456 1.51498084 1.57031808 1.53084828]
Updated Posterior Mean d18Oc Seasonal:
 [2.14406377 2.40704664 2.51005132 2.2920204  1.6697381  1.04506582
 0.43725492 0.43649768 0.73926435 1.04400357 1.51880761 1.79985107]
Updated Posterior Standard Deviation d18Oc Seasonal:
 [0.32353649 0.30673195 0.28355486 0.26365731 0.29381039 0.33005044
 0.34958791 0.37555598 0.36170544 0.31439898 0.33013142 0.33440287]
Original Prior Mean precipitation Seasonal:
 [3.35899606 2.79623688 2.18241293 1.75411782 1.52990136 1.37490782
 1.44029882 2.14277374 2.72446826 3.26583779 3.82762136 3.79819106]
Original Prior Standard Deviation precipitation Seasonal:
 [0.43419051 0.4450087  0.45465416 0.31157176 0.20340219 0.28421396
 0.4935758  0.5372476  0.459027   0.3647094  0.31141638 0.5035945 ]
Updated Posterior Mean precipitation Seasonal:
 [4.41922427 4.012206   3.45068477 2.17710821 1.68089734 1.57664218
 2.22759417 2.59459649 2.9418718  4.15445775 4.51141678 5.16670794]
Updated Posterior Standard Deviation precipitation Seasonal:
 [0.34843545 0.37530378 0.36024202 0.15887241 0.10853529 0.25115884
 0.51632387 0.59186486 0.45682615 0.33330924 0.20506994 0.42471078]

# --- D47 ---

# Plot the updated seasonal posterior for SST_D47
std_post_SST_D47_monthly_aggregated_Railsback_green = np.sqrt(np.diag(cov_post_SST_D47_monthly_aggregated_Railsback_green))
std_prior_SST_D47_monthly_original = np.sqrt(np.diag(cov_prior_SST_D47_monthly_original))
var_start_D47_monthly = var_names.index("D47_corrETF_strict") * num_bins_monthly # Determine the start index for the D47 variable
var_end_D47_monthly = var_start_D47_monthly + num_bins_monthly # Determine the end index for the D47 variable
n_update_monthly_aggregated_Railsback_green_D47 = n_update_monthly_aggregated_Railsback_green[var_start_D47_monthly:var_end_D47_monthly]  # Extract the number of updates for D47

plt.figure(figsize=(10, 6))

# PRIOR
plt.plot(months_scale, mu_prior_SST_D47_monthly_original, label='Prior Mean (PlioMIP models)', color='b', marker='o')
plt.fill_between(
    months_scale,
    mu_prior_SST_D47_monthly_original - stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_SST_D47_monthly_original / np.sqrt(n_models_monthly),
    mu_prior_SST_D47_monthly_original + stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_SST_D47_monthly_original / np.sqrt(n_models_monthly),
    color='b',
    alpha=0.2,
    label='95% Confidence Interval'
)

# LIKELIHOOD
# Determine the start and end indices for the selected variable to parse information from the likelihood statistics
plt.plot(months_scale, mu_likelihood_monthly_aggregated_Railsback_green[var_start_D47_monthly:var_end_D47_monthly], label='Likelihood Mean (clumped data)', color='y', marker='o')
plt.fill_between(
    months_scale,
    mu_likelihood_monthly_aggregated_Railsback_green[var_start_D47_monthly:var_end_D47_monthly] - stats.t.ppf(1 - 0.025, n_update_monthly_aggregated_Railsback_green_D47) * std_likelihood_monthly_aggregated_Railsback_green[var_start_D47_monthly:var_end_D47_monthly] / np.sqrt(n_update_monthly_aggregated_Railsback_green_D47),
    mu_likelihood_monthly_aggregated_Railsback_green[var_start_D47_monthly:var_end_D47_monthly] + stats.t.ppf(1 - 0.025, n_update_monthly_aggregated_Railsback_green_D47) * std_likelihood_monthly_aggregated_Railsback_green[var_start_D47_monthly:var_end_D47_monthly] / np.sqrt(n_update_monthly_aggregated_Railsback_green_D47),
    color='y',
    alpha=0.2,
    label='95% Confidence Interval'
)
for measurement in PWP_data_dict:
    plt.plot(int(measurement["month_score2"]) + 1, measurement["D47_corrETF_strict"], color="y", marker="o", alpha=0.2)
    plt.errorbar(int(measurement["month_score2"]) + 1, measurement["D47_corrETF_strict"], yerr=measurement["D47_sd_ext"], color="y", alpha=0.2, capsize=5)
secax = plt.gca().secondary_xaxis('top')
secax.set_xticks(months_scale)
secax.set_xticklabels([f"n = {int(n)}" for n in n_update_monthly_aggregated_Railsback_green_D47])
secax.tick_params(axis='x', rotation=0)

# POSTERIOR
plt.plot(months_scale, mu_post_SST_D47_monthly_aggregated_Railsback_green, label='Posterior Mean (PlioMIP models + clumped data)', color='r', marker='o')
plt.fill_between(
    months_scale,
    mu_post_SST_D47_monthly_aggregated_Railsback_green - stats.t.ppf(1 - 0.025, (n_update_monthly_aggregated_Railsback_green_D47 + n_models_monthly)) * std_post_SST_D47_monthly_aggregated_Railsback_green / np.sqrt(n_update_monthly_aggregated_Railsback_green_D47 + n_models_monthly),
    mu_post_SST_D47_monthly_aggregated_Railsback_green + stats.t.ppf(1 - 0.025, (n_update_monthly_aggregated_Railsback_green_D47 + n_models_monthly)) * std_post_SST_D47_monthly_aggregated_Railsback_green / np.sqrt(n_update_monthly_aggregated_Railsback_green_D47 + n_models_monthly),
    color='r',
    alpha=0.2,
    label='95% Confidence Interval (Posterior)'
)

plt.xticks(months_scale, month_names, rotation=45, ha="right")
plt.title('Prior and posterior Mean and 95% Confidence Interval for Seasonal SST D47 values\n(Based on individual measurements)')
plt.xlabel('Month')
plt.ylabel('SST D47 value')
plt.legend()
plt.grid(True)
plt.show()

# ---d18Oc---

# Plot the updated seasonal posterior for d18Oc
std_post_d18Oc_monthly_aggregated_Railsback_green = np.sqrt(np.diag(cov_post_d18Oc_Railsback_green_monthly_aggregated))
std_prior_d18Oc_Railsback_green_monthly_original = np.sqrt(np.diag(cov_prior_d18Oc_Railsback_green_monthly_original))
var_start_d18Oc_monthly = var_names.index("Final_d18O") * num_bins_monthly # Determine the start index for the d18Oc variable
var_end_d18Oc_monthly = var_start_d18Oc_monthly + num_bins_monthly # Determine the end index for the d18Oc variable
n_update_monthly_aggregated_Railsback_green_d18Oc = n_update_monthly_aggregated_Railsback_green[var_start_d18Oc_monthly:var_end_d18Oc_monthly]  # Extract the number of updates for d18Oc

plt.figure(figsize=(10, 6))

# PRIOR
plt.plot(months_scale, mu_prior_d18Oc_Railsback_green_monthly_original, label='Prior Mean (PlioMIP models)', color='b', marker='o')
plt.fill_between(
    months_scale,
    mu_prior_d18Oc_Railsback_green_monthly_original - stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_d18Oc_Railsback_green_monthly_original / np.sqrt(n_models_monthly),
    mu_prior_d18Oc_Railsback_green_monthly_original + stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_d18Oc_Railsback_green_monthly_original / np.sqrt(n_models_monthly),
    color='b',
    alpha=0.2,
    label='95% Confidence Interval'
)

# LIKELIHOOD
# Determine the start and end indices for the selected variable
plt.plot(months_scale, mu_likelihood_monthly_aggregated_Railsback_green[var_start_d18Oc_monthly:var_end_d18Oc_monthly], label='Likelihood Mean (clumped data)', color='y', marker='o')
plt.fill_between(
    months_scale,
    mu_likelihood_monthly_aggregated_Railsback_green[var_start_d18Oc_monthly:var_end_d18Oc_monthly] - stats.t.ppf(1 - 0.025, n_update_monthly_aggregated_Railsback_green_d18Oc) * std_likelihood_monthly_aggregated_Railsback_green[var_start_d18Oc_monthly:var_end_d18Oc_monthly] / np.sqrt(n_update_monthly_aggregated_Railsback_green_d18Oc),
    mu_likelihood_monthly_aggregated_Railsback_green[var_start_d18Oc_monthly:var_end_d18Oc_monthly] + stats.t.ppf(1 - 0.025, n_update_monthly_aggregated_Railsback_green_d18Oc) * std_likelihood_monthly_aggregated_Railsback_green[var_start_d18Oc_monthly:var_end_d18Oc_monthly] / np.sqrt(n_update_monthly_aggregated_Railsback_green_d18Oc),
    color='y',
    alpha=0.2,
    label='95% Confidence Interval'
)
for measurement in PWP_data_dict:
    plt.plot(int(measurement["month_score2"]) + 1, measurement["Final_d18O"], color="y", marker="o", alpha=0.2)
    plt.errorbar(int(measurement["month_score2"]) + 1, measurement["Final_d18O"], yerr=measurement["d18O_sd_ext"], color="y", alpha=0.2, capsize=5)
secax = plt.gca().secondary_xaxis('top')
secax.set_xticks(months_scale)
secax.set_xticklabels([f"n = {int(n)}" for n in n_update_monthly_aggregated_Railsback_green_d18Oc])
secax.tick_params(axis='x', rotation=0)

# POSTERIOR
plt.plot(months_scale, mu_post_d18Oc_Railsback_green_monthly_aggregated, label='Posterior Mean (PlioMIP models + clumped data)', color='r', marker='o')
plt.fill_between(
    months_scale,
    mu_post_d18Oc_Railsback_green_monthly_aggregated - stats.t.ppf(1 - 0.025, (n_update_monthly_aggregated_Railsback_green_d18Oc + n_models_monthly)) * std_post_d18Oc_monthly_aggregated_Railsback_green / np.sqrt(n_update_monthly_aggregated_Railsback_green_d18Oc + n_models_monthly),
    mu_post_d18Oc_Railsback_green_monthly_aggregated + stats.t.ppf(1 - 0.025, (n_update_monthly_aggregated_Railsback_green_d18Oc + n_models_monthly)) * std_post_d18Oc_monthly_aggregated_Railsback_green / np.sqrt(n_update_monthly_aggregated_Railsback_green_d18Oc + n_models_monthly),
    color='r',
    alpha=0.2,
    label='95% Confidence Interval (Posterior)'
)

plt.xticks(months_scale, month_names, rotation=45, ha="right")
plt.title('Prior and posterior Mean and 95% Confidence Interval for Seasonal d18Oc values\n(Based on individual measurements)')
plt.xlabel('Month')
plt.ylabel('d18Oc value')
plt.legend()
plt.grid(True)
plt.show()

# Function to propagate covariance on d18Ow in case of linear d18Oc-d18Ow-T equation
def propagate_cov_d18Ow_linear(cov_c, cov_T, cov_cT, A):
    """
    Propagate covariance for:
        w = c - (B - T)/A (+ 0.27)
    where A and B are scalars and the term 0.27 is used to convert between SMOW and VSMOW, but only A is relevant for covariance

    Inputs:
        cov_c  : (n,n) covariance matrix of d18Oc
        cov_T  : (n,n) covariance matrix of T
        cov_cT : (n,n) cross-covariance matrix Cov(c,T)
        A      : scalar (slope)

    Returns:
        cov_w : (n,n) covariance matrix of d18Ow
    """

    dwdc = 1.0
    dwdT = 1.0 / A

    cov_w = (
        dwdc**2 * cov_c
        + dwdc * dwdT * (cov_cT + cov_cT.T)
        + dwdT**2 * cov_T
    )

    return cov_w

# PRIORS
# Convert prior D47 to temp
mu_prior_SST_D47_monthly_T = D47c.OGLS23.T47(D47 = mu_prior_SST_D47_monthly_original, sD47 = cov_prior_SST_D47_monthly_original, return_covar = True)[0]
cov_prior_SST_D47_monthly_T = D47c.OGLS23.T47(D47 = mu_prior_SST_D47_monthly_original, sD47 = cov_prior_SST_D47_monthly_original, return_covar = True)[1]

# Convert prior d18Oc and temperature to d18Ow using Grossman and Ku (1986) with the VPDB-VSMOW scale correction by Gonfiantini et al. (1995) and Dettman et al. (1999)
mu_prior_d18Ow_Harwood_monthly_T = mu_prior_d18Oc_Harwood_monthly_original - (20.6 - mu_prior_SST_D47_monthly_T) / 4.34 + 0.27
mu_prior_d18Ow_Graniero_monthly_T = mu_prior_d18Oc_Graniero_monthly_original - (20.6 - mu_prior_SST_D47_monthly_T) / 4.34 + 0.27
mu_prior_d18Ow_Railsback_ice_monthly_T = mu_prior_d18Oc_Railsback_ice_monthly_original - (20.6 - mu_prior_SST_D47_monthly_T) / 4.34 + 0.27
mu_prior_d18Ow_Railsback_green_monthly_T = mu_prior_d18Oc_Railsback_green_monthly_original - (20.6 - mu_prior_SST_D47_monthly_T) / 4.34 + 0.27
# Calculate cross covariance between d18Oc and temperature for each dataset
cross_cov_prior_d18Oc_SST_monthly_Harwood = np.cov(PlioMIP[SST_D47_columns_monthly + d18Oc_Harwood_columns_monthly].dropna(), rowvar=False)[len(SST_D47_columns_monthly):, :len(SST_D47_columns_monthly)]
cross_cov_prior_d18Oc_SST_monthly_Graniero = np.cov(PlioMIP[SST_D47_columns_monthly + d18Oc_Graniero_columns_monthly].dropna(), rowvar=False)[len(SST_D47_columns_monthly):, :len(SST_D47_columns_monthly)]
cross_cov_prior_d18Oc_SST_monthly_Railsback_ice = np.cov(PlioMIP[SST_D47_columns_monthly + d18Oc_Railsback_ice_columns_monthly].dropna(), rowvar=False)[len(SST_D47_columns_monthly):, :len(SST_D47_columns_monthly)]
cross_cov_prior_d18Oc_SST_monthly_Railsback_green = np.cov(PlioMIP[SST_D47_columns_monthly + d18Oc_Railsback_green_columns_monthly].dropna(), rowvar=False)[len(SST_D47_columns_monthly):, :len(SST_D47_columns_monthly)]
# Propagate covariance to d18Ow for each dataset
cov_prior_d18Ow_Harwood_monthly_T = propagate_cov_d18Ow_linear(
    cov_prior_d18Oc_Harwood_monthly_original,
    cov_prior_SST_D47_monthly_T,
    cross_cov_prior_d18Oc_SST_monthly_Harwood,
    4.34
)
cov_prior_d18Ow_Graniero_monthly_T = propagate_cov_d18Ow_linear(
    cov_prior_d18Oc_Graniero_monthly_original,
    cov_prior_SST_D47_monthly_T,
    cross_cov_prior_d18Oc_SST_monthly_Graniero,
    4.34
)
cov_prior_d18Ow_Railsback_ice_monthly_T = propagate_cov_d18Ow_linear(
    cov_prior_d18Oc_Railsback_ice_monthly_original,
    cov_prior_SST_D47_monthly_T,
    cross_cov_prior_d18Oc_SST_monthly_Railsback_ice,
    4.34
)
cov_prior_d18Ow_Railsback_green_monthly_T = propagate_cov_d18Ow_linear(
    cov_prior_d18Oc_Railsback_green_monthly_original,
    cov_prior_SST_D47_monthly_T,
    cross_cov_prior_d18Oc_SST_monthly_Railsback_green,
    4.34
)

# Convert prior d18Ow to SSS using different conversion functions
mu_prior_SSS_d18Ow_Harwood_monthly_T = (mu_prior_d18Ow_Harwood_monthly_T + 9.300) / 0.274
mu_prior_SSS_d18Ow_Graniero_monthly_T = (mu_prior_d18Ow_Graniero_monthly_T + 7.89) / 0.23
mu_prior_SSS_d18Ow_Railsback_ice_monthly_T = mu_prior_d18Ow_Railsback_ice_monthly_T * 35 / 21 + 35
mu_prior_SSS_d18Ow_Railsback_green_monthly_T = (mu_prior_d18Ow_Railsback_green_monthly_T + 1.08) * 35 / 8 + 35
cov_prior_SSS_d18Ow_Harwood_monthly_T = cov_prior_d18Ow_Harwood_monthly_T / (0.274 ** 2)
cov_prior_SSS_d18Ow_Graniero_monthly_T = cov_prior_d18Ow_Graniero_monthly_T / (0.23 ** 2)
cov_prior_SSS_d18Ow_Railsback_ice_monthly_T = cov_prior_d18Ow_Railsback_ice_monthly_T / (35 / 21) ** 2
cov_prior_SSS_d18Ow_Railsback_green_monthly_T = cov_prior_d18Ow_Railsback_green_monthly_T / (35 / 8) ** 2

# LIKELIHOODS
# Convert likelihood D47 to temp
mu_likelihood_monthly_aggregated_Harwood_T = D47c.OGLS23.T47(D47 = mu_likelihood_monthly_aggregated_Harwood[var_start_D47_monthly:var_end_D47_monthly], sD47 = std_likelihood_monthly_aggregated_Harwood[var_start_D47_monthly:var_end_D47_monthly], return_covar = True)[0]
mu_likelihood_monthly_aggregated_Graniero_T = D47c.OGLS23.T47(D47 = mu_likelihood_monthly_aggregated_Graniero[var_start_D47_monthly:var_end_D47_monthly], sD47 = std_likelihood_monthly_aggregated_Graniero[var_start_D47_monthly:var_end_D47_monthly], return_covar = True)[0]
mu_likelihood_monthly_aggregated_Railsback_ice_T = D47c.OGLS23.T47(D47 = mu_likelihood_monthly_aggregated_Railsback_ice[var_start_D47_monthly:var_end_D47_monthly], sD47 = std_likelihood_monthly_aggregated_Railsback_ice[var_start_D47_monthly:var_end_D47_monthly], return_covar = True)[0]
mu_likelihood_monthly_aggregated_Railsback_green_T = D47c.OGLS23.T47(D47 = mu_likelihood_monthly_aggregated_Railsback_green[var_start_D47_monthly:var_end_D47_monthly], sD47 = std_likelihood_monthly_aggregated_Railsback_green[var_start_D47_monthly:var_end_D47_monthly], return_covar = True)[0]
cov_likelihood_monthly_aggregated_Harwood_T = D47c.OGLS23.T47(D47 = mu_likelihood_monthly_aggregated_Harwood[var_start_D47_monthly:var_end_D47_monthly], sD47 = std_likelihood_monthly_aggregated_Harwood[var_start_D47_monthly:var_end_D47_monthly], return_covar = True)[1]
cov_likelihood_monthly_aggregated_Graniero_T = D47c.OGLS23.T47(D47 = mu_likelihood_monthly_aggregated_Graniero[var_start_D47_monthly:var_end_D47_monthly], sD47 = std_likelihood_monthly_aggregated_Graniero[var_start_D47_monthly:var_end_D47_monthly], return_covar = True)[1]
cov_likelihood_monthly_aggregated_Railsback_ice_T = D47c.OGLS23.T47(D47 = mu_likelihood_monthly_aggregated_Railsback_ice[var_start_D47_monthly:var_end_D47_monthly], sD47 = std_likelihood_monthly_aggregated_Railsback_ice[var_start_D47_monthly:var_end_D47_monthly], return_covar = True)[1]
cov_likelihood_monthly_aggregated_Railsback_green_T = D47c.OGLS23.T47(D47 = mu_likelihood_monthly_aggregated_Railsback_green[var_start_D47_monthly:var_end_D47_monthly], sD47 = std_likelihood_monthly_aggregated_Railsback_green[var_start_D47_monthly:var_end_D47_monthly], return_covar = True)[1]

# Convert likelihood d18Oc and temperature to d18Ow
mu_likelihood_d18Ow_monthly_aggregated_Harwood_T = mu_likelihood_monthly_aggregated_Harwood[var_start_d18Oc_monthly:var_end_d18Oc_monthly] - (20.6 - mu_likelihood_monthly_aggregated_Harwood_T) / 4.34 + 0.27
mu_likelihood_d18Ow_monthly_aggregated_Graniero_T = mu_likelihood_monthly_aggregated_Graniero[var_start_d18Oc_monthly:var_end_d18Oc_monthly] - (20.6 - mu_likelihood_monthly_aggregated_Graniero_T) / 4.34 + 0.27
mu_likelihood_d18Ow_monthly_aggregated_Railsback_ice_T = mu_likelihood_monthly_aggregated_Railsback_ice[var_start_d18Oc_monthly:var_end_d18Oc_monthly] - (20.6 - mu_likelihood_monthly_aggregated_Railsback_ice_T) / 4.34 + 0.27
mu_likelihood_d18Ow_monthly_aggregated_Railsback_green_T = mu_likelihood_monthly_aggregated_Railsback_green[var_start_d18Oc_monthly:var_end_d18Oc_monthly] - (20.6 - mu_likelihood_monthly_aggregated_Railsback_green_T) / 4.34 + 0.27
# Use covariance propagation to calculate covariance on d18Ow for each dataset
std_likelihood_d18Ow_monthly_aggregated_Harwood_T = np.sqrt(
    np.diag(
        propagate_cov_d18Ow_linear(
            np.diag(std_likelihood_monthly_aggregated_Harwood[var_start_d18Oc_monthly:var_end_d18Oc_monthly]**2),
            cov_likelihood_monthly_aggregated_Harwood_T,
            cross_cov_prior_d18Oc_SST_monthly_Harwood,
            4.34
        )
    )
)
std_likelihood_d18Ow_monthly_aggregated_Graniero_T = np.sqrt(
    np.diag(
        propagate_cov_d18Ow_linear(
            np.diag(std_likelihood_monthly_aggregated_Graniero[var_start_d18Oc_monthly:var_end_d18Oc_monthly]**2),
            cov_likelihood_monthly_aggregated_Graniero_T,
            cross_cov_prior_d18Oc_SST_monthly_Graniero,
            4.34
        )
    )
)
std_likelihood_d18Ow_monthly_aggregated_Railsback_ice_T = np.sqrt(
    np.diag(
        propagate_cov_d18Ow_linear(
            np.diag(std_likelihood_monthly_aggregated_Railsback_ice[var_start_d18Oc_monthly:var_end_d18Oc_monthly]**2),
            cov_likelihood_monthly_aggregated_Railsback_ice_T,
            cross_cov_prior_d18Oc_SST_monthly_Railsback_ice,
            4.34
        )
    )
)
std_likelihood_d18Ow_monthly_aggregated_Railsback_green_T = np.sqrt(
    np.diag(
        propagate_cov_d18Ow_linear(
            np.diag(std_likelihood_monthly_aggregated_Railsback_green[var_start_d18Oc_monthly:var_end_d18Oc_monthly]**2),
            cov_likelihood_monthly_aggregated_Railsback_green_T,
            cross_cov_prior_d18Oc_SST_monthly_Railsback_green,
            4.34
        )
    )
)

mu_likelihood_SSS_d18Ow_monthly_aggregated_Harwood_T = (mu_likelihood_d18Ow_monthly_aggregated_Harwood_T + 9.300) / 0.274
mu_likelihood_SSS_d18Ow_monthly_aggregated_Graniero_T = (mu_likelihood_d18Ow_monthly_aggregated_Graniero_T + 7.89) / 0.23
mu_likelihood_SSS_d18Ow_monthly_aggregated_Railsback_ice_T = mu_likelihood_d18Ow_monthly_aggregated_Railsback_ice_T * 35 / 21 + 35
mu_likelihood_SSS_d18Ow_monthly_aggregated_Railsback_green_T = (mu_likelihood_d18Ow_monthly_aggregated_Railsback_green_T + 1.08) * 35 / 8 + 35
std_likelihood_SSS_d18Ow_monthly_aggregated_Harwood_T = std_likelihood_d18Ow_monthly_aggregated_Harwood_T / (0.274 ** 2)
std_likelihood_SSS_d18Ow_monthly_aggregated_Graniero_T = std_likelihood_d18Ow_monthly_aggregated_Graniero_T / (0.23 ** 2)
std_likelihood_SSS_d18Ow_monthly_aggregated_Railsback_ice_T = std_likelihood_d18Ow_monthly_aggregated_Railsback_ice_T / (35 / 21) ** 2
std_likelihood_SSS_d18Ow_monthly_aggregated_Railsback_green_T = std_likelihood_d18Ow_monthly_aggregated_Railsback_green_T / (35 / 8) ** 2

# POSTERIORS
# Convert posterior SST-D47 to temp
mu_post_SST_D47_monthly_aggregated_Harwood_T = D47c.OGLS23.T47(D47 = mu_post_SST_D47_monthly_aggregated_Harwood, sD47 = cov_post_SST_D47_monthly_aggregated_Harwood, return_covar = True)[0] 
mu_post_SST_D47_monthly_aggregated_Graniero_T = D47c.OGLS23.T47(D47 = mu_post_SST_D47_monthly_aggregated_Graniero, sD47 = cov_post_SST_D47_monthly_aggregated_Graniero, return_covar = True)[0]
mu_post_SST_D47_monthly_aggregated_Railsback_ice_T = D47c.OGLS23.T47(D47 = mu_post_SST_D47_monthly_aggregated_Railsback_ice, sD47 = cov_post_SST_D47_monthly_aggregated_Railsback_ice, return_covar = True)[0]
mu_post_SST_D47_monthly_aggregated_Railsback_green_T = D47c.OGLS23.T47(D47 = mu_post_SST_D47_monthly_aggregated_Railsback_green, sD47 = cov_post_SST_D47_monthly_aggregated_Railsback_green, return_covar = True)[0]
cov_post_SST_D47_monthly_aggregated_Harwood_T = D47c.OGLS23.T47(D47 = mu_post_SST_D47_monthly_aggregated_Harwood, sD47 = cov_post_SST_D47_monthly_aggregated_Harwood, return_covar = True)[1]
cov_post_SST_D47_monthly_aggregated_Graniero_T = D47c.OGLS23.T47(D47 = mu_post_SST_D47_monthly_aggregated_Graniero, sD47 = cov_post_SST_D47_monthly_aggregated_Graniero, return_covar = True)[1]
cov_post_SST_D47_monthly_aggregated_Railsback_ice_T = D47c.OGLS23.T47(D47 = mu_post_SST_D47_monthly_aggregated_Railsback_ice, sD47 = cov_post_SST_D47_monthly_aggregated_Railsback_ice, return_covar = True)[1]
cov_post_SST_D47_monthly_aggregated_Railsback_green_T = D47c.OGLS23.T47(D47 = mu_post_SST_D47_monthly_aggregated_Railsback_green, sD47 = cov_post_SST_D47_monthly_aggregated_Railsback_green, return_covar = True)[1]

# Convert posterior SAT-D47 to temp
mu_post_SAT_D47_monthly_aggregated_Harwood_T = D47c.OGLS23.T47(D47 = mu_post_SAT_D47_monthly_aggregated_Harwood, sD47 = cov_post_SAT_D47_monthly_aggregated_Harwood, return_covar = True)[0] 
mu_post_SAT_D47_monthly_aggregated_Graniero_T = D47c.OGLS23.T47(D47 = mu_post_SAT_D47_monthly_aggregated_Graniero, sD47 = cov_post_SAT_D47_monthly_aggregated_Graniero, return_covar = True)[0]
mu_post_SAT_D47_monthly_aggregated_Railsback_ice_T = D47c.OGLS23.T47(D47 = mu_post_SAT_D47_monthly_aggregated_Railsback_ice, sD47 = cov_post_SAT_D47_monthly_aggregated_Railsback_ice, return_covar = True)[0]
mu_post_SAT_D47_monthly_aggregated_Railsback_green_T = D47c.OGLS23.T47(D47 = mu_post_SAT_D47_monthly_aggregated_Railsback_green, sD47 = cov_post_SAT_D47_monthly_aggregated_Railsback_green, return_covar = True)[0]
cov_post_SAT_D47_monthly_aggregated_Harwood_T = D47c.OGLS23.T47(D47 = mu_post_SAT_D47_monthly_aggregated_Harwood, sD47 = cov_post_SAT_D47_monthly_aggregated_Harwood, return_covar = True)[1]
cov_post_SAT_D47_monthly_aggregated_Graniero_T = D47c.OGLS23.T47(D47 = mu_post_SAT_D47_monthly_aggregated_Graniero, sD47 = cov_post_SAT_D47_monthly_aggregated_Graniero, return_covar = True)[1]
cov_post_SAT_D47_monthly_aggregated_Railsback_ice_T = D47c.OGLS23.T47(D47 = mu_post_SAT_D47_monthly_aggregated_Railsback_ice, sD47 = cov_post_SAT_D47_monthly_aggregated_Railsback_ice, return_covar = True)[1]
cov_post_SAT_D47_monthly_aggregated_Railsback_green_T = D47c.OGLS23.T47(D47 = mu_post_SAT_D47_monthly_aggregated_Railsback_green, sD47 = cov_post_SAT_D47_monthly_aggregated_Railsback_green, return_covar = True)[1]

# Convert posterior d18Oc and temperature to d18Ow
mu_post_d18Ow_monthly_aggregated_Harwood_T = mu_post_d18Oc_Harwood_monthly_aggregated - (20.6 - mu_post_SST_D47_monthly_aggregated_Harwood_T) / 4.34 + 0.27
mu_post_d18Ow_monthly_aggregated_Graniero_T = mu_post_d18Oc_Graniero_monthly_aggregated - (20.6 - mu_post_SST_D47_monthly_aggregated_Graniero_T) / 4.34 + 0.27
mu_post_d18Ow_monthly_aggregated_Railsback_ice_T = mu_post_d18Oc_Railsback_ice_monthly_aggregated - (20.6 - mu_post_SST_D47_monthly_aggregated_Railsback_ice_T) / 4.34 + 0.27
mu_post_d18Ow_monthly_aggregated_Railsback_green_T = mu_post_d18Oc_Railsback_green_monthly_aggregated - (20.6 - mu_post_SST_D47_monthly_aggregated_Railsback_green_T) / 4.34 + 0.27
# Use propagation function to calculate covariance on d18Ow for each dataset
cov_post_d18Ow_monthly_aggregated_Harwood_T = propagate_cov_d18Ow_linear(
    cov_post_d18Oc_Harwood_monthly_aggregated,
    cov_post_SST_D47_monthly_aggregated_Harwood_T,
    cross_cov_prior_d18Oc_SST_monthly_Harwood,
    4.34
)
cov_post_d18Ow_monthly_aggregated_Graniero_T = propagate_cov_d18Ow_linear(
    cov_post_d18Oc_Graniero_monthly_aggregated,
    cov_post_SST_D47_monthly_aggregated_Graniero_T,
    cross_cov_prior_d18Oc_SST_monthly_Graniero,
    4.34
)
cov_post_d18Ow_monthly_aggregated_Railsback_ice_T = propagate_cov_d18Ow_linear(
    cov_post_d18Oc_Railsback_ice_monthly_aggregated,
    cov_post_SST_D47_monthly_aggregated_Railsback_ice_T,
    cross_cov_prior_d18Oc_SST_monthly_Railsback_ice,
    4.34
)
cov_post_d18Ow_monthly_aggregated_Railsback_green_T = propagate_cov_d18Ow_linear(
    cov_post_d18Oc_Railsback_green_monthly_aggregated,
    cov_post_SST_D47_monthly_aggregated_Railsback_green_T,
    cross_cov_prior_d18Oc_SST_monthly_Railsback_green,
    4.34
)

# Convert posterior d18Ow to SSS
mu_post_SSS_d18Ow_monthly_aggregated_Harwood_T = (mu_post_d18Ow_monthly_aggregated_Harwood_T + 9.300) / 0.274
mu_post_SSS_d18Ow_monthly_aggregated_Graniero_T = (mu_post_d18Ow_monthly_aggregated_Graniero_T + 7.89) / 0.23
mu_post_SSS_d18Ow_monthly_aggregated_Railsback_ice_T = mu_post_d18Ow_monthly_aggregated_Railsback_ice_T * 35 / 21 + 35
mu_post_SSS_d18Ow_monthly_aggregated_Railsback_green_T = (mu_post_d18Ow_monthly_aggregated_Railsback_green_T + 1.08) * 35 / 8 + 35
cov_post_SSS_d18Ow_monthly_aggregated_Harwood_T = cov_post_d18Ow_monthly_aggregated_Harwood_T / (0.274 ** 2)
cov_post_SSS_d18Ow_monthly_aggregated_Graniero_T = cov_post_d18Ow_monthly_aggregated_Graniero_T / (0.23 ** 2)
cov_post_SSS_d18Ow_monthly_aggregated_Railsback_ice_T = cov_post_d18Ow_monthly_aggregated_Railsback_ice_T / (35 / 21) ** 2
cov_post_SSS_d18Ow_monthly_aggregated_Railsback_green_T = cov_post_d18Ow_monthly_aggregated_Railsback_green_T / (35 / 8) ** 2

# Calculate standard deviations for prior, likelihood, and posterior in temperature and d18Ow space
std_prior_SST_D47_monthly_T = np.sqrt(np.diag(cov_prior_SST_D47_monthly_T))

std_prior_SSS_d18Ow_Harwood_monthly_T = np.sqrt(np.diag(cov_prior_SSS_d18Ow_Harwood_monthly_T))
std_prior_SSS_d18Ow_Graniero_monthly_T = np.sqrt(np.diag(cov_prior_SSS_d18Ow_Graniero_monthly_T))
std_prior_SSS_d18Ow_Railsback_ice_monthly_T = np.sqrt(np.diag(cov_prior_SSS_d18Ow_Railsback_ice_monthly_T))
std_prior_SSS_d18Ow_Railsback_green_monthly_T = np.sqrt(np.diag(cov_prior_SSS_d18Ow_Railsback_green_monthly_T))

std_likelihood_monthly_aggregated_Harwood_T = np.sqrt(np.diag(cov_likelihood_monthly_aggregated_Harwood_T))
std_likelihood_monthly_aggregated_Graniero_T = np.sqrt(np.diag(cov_likelihood_monthly_aggregated_Graniero_T))
std_likelihood_monthly_aggregated_Railsback_ice_T = np.sqrt(np.diag(cov_likelihood_monthly_aggregated_Railsback_ice_T))
std_likelihood_monthly_aggregated_Railsback_green_T = np.sqrt(np.diag(cov_likelihood_monthly_aggregated_Railsback_green_T))

std_post_SST_D47_monthly_aggregated_Harwood_T = np.sqrt(np.diag(cov_post_SST_D47_monthly_aggregated_Harwood_T))
std_post_SST_D47_monthly_aggregated_Graniero_T = np.sqrt(np.diag(cov_post_SST_D47_monthly_aggregated_Graniero_T))
std_post_SST_D47_monthly_aggregated_Railsback_ice_T = np.sqrt(np.diag(cov_post_SST_D47_monthly_aggregated_Railsback_ice_T))
std_post_SST_D47_monthly_aggregated_Railsback_green_T = np.sqrt(np.diag(cov_post_SST_D47_monthly_aggregated_Railsback_green_T))

std_post_SAT_D47_monthly_aggregated_Harwood_T = np.sqrt(np.diag(cov_post_SAT_D47_monthly_aggregated_Harwood_T))
std_post_SAT_D47_monthly_aggregated_Graniero_T = np.sqrt(np.diag(cov_post_SAT_D47_monthly_aggregated_Graniero_T))
std_post_SAT_D47_monthly_aggregated_Railsback_ice_T = np.sqrt(np.diag(cov_post_SAT_D47_monthly_aggregated_Railsback_ice_T))
std_post_SAT_D47_monthly_aggregated_Railsback_green_T = np.sqrt(np.diag(cov_post_SAT_D47_monthly_aggregated_Railsback_green_T))

std_post_SSS_d18Ow_monthly_aggregated_Harwood_T = np.sqrt(np.diag(cov_post_SSS_d18Ow_monthly_aggregated_Harwood_T))
std_post_SSS_d18Ow_monthly_aggregated_Graniero_T = np.sqrt(np.diag(cov_post_SSS_d18Ow_monthly_aggregated_Graniero_T))
std_post_SSS_d18Ow_monthly_aggregated_Railsback_ice_T = np.sqrt(np.diag(cov_post_SSS_d18Ow_monthly_aggregated_Railsback_ice_T))
std_post_SSS_d18Ow_monthly_aggregated_Railsback_green_T = np.sqrt(np.diag(cov_post_SSS_d18Ow_monthly_aggregated_Railsback_green_T))

# # Initiate plot
# fig, axs = plt.subplots(2, 1, figsize=(10, 12), sharex=True)

# # FIRST PANEL: SST Results
# # PRIOR
# axs[0].plot(months_scale, mu_prior_SST_D47_monthly_T, label='Prior Mean (PlioMIP models)', color='b', marker='o')
# axs[0].fill_between(months_scale,
#     mu_prior_SST_D47_monthly_T - stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_SST_D47_monthly_T / np.sqrt(n_models_monthly),
#     mu_prior_SST_D47_monthly_T + stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_SST_D47_monthly_T / np.sqrt(n_models_monthly),
#     color='b', alpha=0.2, label='95% Confidence Interval')

# # LIKELIHOOD
# axs[0].plot(months_scale, mu_likelihood_monthly_aggregated_Harwood_T, label='Likelihood Mean (clumped data)', color='y', marker='o')
# axs[0].fill_between(months_scale,
#     mu_likelihood_monthly_aggregated_Harwood_T - stats.t.ppf(1 - 0.025, n_update_monthly_aggregated_Harwood_D47) * std_likelihood_monthly_aggregated_Harwood_T / np.sqrt(n_update_monthly_aggregated_Harwood_D47),
#     mu_likelihood_monthly_aggregated_Harwood_T + stats.t.ppf(1 - 0.025, n_update_monthly_aggregated_Harwood_D47) * std_likelihood_monthly_aggregated_Harwood_T / np.sqrt(n_update_monthly_aggregated_Harwood_D47),
#     color='y', alpha=0.2, label='95% Confidence Interval')

# # POSTERIOR
# axs[0].plot(months_scale, mu_post_SST_D47_monthly_aggregated_Harwood_T, label='Posterior Mean (PlioMIP models + clumped data)', color='r', marker='o')
# axs[0].fill_between(months_scale,
#     mu_post_SST_D47_monthly_aggregated_Harwood_T - stats.t.ppf(1 - 0.025, (n_update_monthly_aggregated_Harwood_D47 + n_models_monthly)) * std_post_SST_D47_monthly_aggregated_T / np.sqrt(n_update_monthly_aggregated_Harwood_D47 + n_models_monthly),
#     mu_post_SST_D47_monthly_aggregated_Harwood_T + stats.t.ppf(1 - 0.025, (n_update_monthly_aggregated_Harwood_D47 + n_models_monthly)) * std_post_SST_D47_monthly_aggregated_T / np.sqrt(n_update_monthly_aggregated_Harwood_D47 + n_models_monthly),
#     color='r', alpha=0.2, label='95% Confidence Interval (Posterior)')

# # Layout for SST panel
# axs[0].set_title('monthly Sea Surface Temperatures (SST)')
# axs[0].set_ylabel('Temperature (°C)')
# axs[0].legend(loc='upper left')
# axs[0].grid(True)

# # SECOND PANEL: SSS Results
# # PRIOR
# axs[1].plot(months_scale, mu_prior_SSS_d18Ow_Harwood_monthly_T, label='Prior Mean (PlioMIP models)', color='b', marker='o')
# axs[1].fill_between(months_scale,
#     mu_prior_SSS_d18Ow_Harwood_monthly_T - stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_SSS_d18Ow_monthly_T / np.sqrt(n_models_monthly),
#     mu_prior_SSS_d18Ow_Harwood_monthly_T + stats.t.ppf(1 - 0.025, n_models_monthly) * std_prior_SSS_d18Ow_monthly_T / np.sqrt(n_models_monthly),
#     color='b', alpha=0.2, label='95% Confidence Interval')

# # LIKELIHOOD
# axs[1].plot(months_scale, mu_likelihood_SSS_d18Ow_monthly_aggregated_Harwood_T, label='Likelihood Mean (clumped data)', color='y', marker='o')
# axs[1].fill_between(months_scale,
#     mu_likelihood_SSS_d18Ow_monthly_aggregated_Harwood_T - stats.t.ppf(1 - 0.025, n_update_monthly_aggregated_Harwood_d18Oc) * std_likelihood_SSS_d18Ow_monthly_aggregated_Harwood_T / np.sqrt(n_update_monthly_aggregated_Harwood_d18Oc),
#     mu_likelihood_SSS_d18Ow_monthly_aggregated_Harwood_T + stats.t.ppf(1 - 0.025, n_update_monthly_aggregated_Harwood_d18Oc) * std_likelihood_SSS_d18Ow_monthly_aggregated_Harwood_T / np.sqrt(n_update_monthly_aggregated_Harwood_d18Oc),
#     color='y', alpha=0.2, label='95% Confidence Interval')

# # POSTERIOR
# axs[1].plot(months_scale, mu_post_SSS_d18Ow_monthly_aggregated_T, label='Posterior Mean (PlioMIP models + clumped data)', color='r', marker='o')
# axs[1].fill_between(months_scale,
#     mu_post_SSS_d18Ow_monthly_aggregated_T - stats.t.ppf(1 - 0.025, (n_update_monthly_aggregated_Harwood_d18Oc + n_models_monthly)) * std_post_SSS_d18Ow_monthly_aggregated_T / np.sqrt(n_update_monthly_aggregated_Harwood_d18Oc + n_models_monthly),
#     mu_post_SSS_d18Ow_monthly_aggregated_T + stats.t.ppf(1 - 0.025, (n_update_monthly_aggregated_Harwood_d18Oc + n_models_monthly)) * std_post_SSS_d18Ow_monthly_aggregated_T / np.sqrt(n_update_monthly_aggregated_Harwood_d18Oc + n_models_monthly),
#     color='r', alpha=0.2, label='95% Confidence Interval (Posterior)')

# # Layout for SSS panel
# axs[1].set_title('monthly Sea Surface Salinity (SSS)')
# axs[1].set_xlabel('Month')
# axs[1].set_ylabel('Salinity (PSU)')
# axs[1].legend(loc='upper left')
# axs[1].grid(True)

# # Shared x-axis labels
# plt.xticks(months_scale, month_names, rotation=45, ha="right")
# plt.tight_layout()
# plt.show()

# Create a figure with two rows and two columns of subplots
fig, axes = plt.subplots(2, 2, figsize=(18, 12))

# --- SST & SAT Prior and Posterior ---
# Prior SST & SAT
axes[0, 0].plot(months_scale, mu_prior_SST_monthly, label='Prior SST Mean', marker='o')
axes[0, 0].plot(months_scale, mu_prior_SAT_monthly, label='Prior SAT Mean', marker='o')
axes[0, 0].fill_between(
    months_scale,
    mu_prior_SST_monthly - 2 * std_prior_SST_monthly,
    mu_prior_SST_monthly + 2 * std_prior_SST_monthly,
    alpha=0.2, label='+/- 2 standard deviations (SST)'
)
axes[0, 0].fill_between(
    months_scale,
    mu_prior_SAT_monthly - 2 * std_prior_SAT_monthly,
    mu_prior_SAT_monthly + 2 * std_prior_SAT_monthly,
    alpha=0.2, label='+/- 2 standard deviations (SAT)'
)
axes[0, 0].set_xticks(months_scale)
axes[0, 0].set_xticklabels(month_names, rotation=45, ha="right")
axes[0, 0].set_title('Prior Mean and +/- 2 standard deviations for monthly SST & SAT')
axes[0, 0].set_xlabel('Month')
axes[0, 0].set_ylabel('Temperature (°C)')
axes[0, 0].set_ylim(0, 30)
axes[0, 0].legend(loc='upper left')
axes[0, 0].grid(True)

# Posterior SST & SAT
axes[0, 1].plot(months_scale, mu_post_SST_D47_monthly_aggregated_Harwood_T, label='Posterior SST Mean', marker='o')
axes[0, 1].plot(months_scale, mu_post_SAT_D47_monthly_aggregated_Harwood_T, label='Posterior SAT Mean', marker='o')
axes[0, 1].fill_between(
    months_scale,
    mu_post_SST_D47_monthly_aggregated_Harwood_T - 2 * std_post_SST_D47_monthly_aggregated_Harwood_T,
    mu_post_SST_D47_monthly_aggregated_Harwood_T + 2 * std_post_SST_D47_monthly_aggregated_Harwood_T,
    alpha=0.2, label='+/- 2 standard deviations (SST)'
)
axes[0, 1].fill_between(
    months_scale,
    mu_post_SAT_D47_monthly_aggregated_Harwood_T - 2 * std_post_SAT_D47_monthly_aggregated_Harwood_T,
    mu_post_SAT_D47_monthly_aggregated_Harwood_T + 2 * std_post_SAT_D47_monthly_aggregated_Harwood_T,
    alpha=0.2, label='+/- 2 standard deviations (SAT)'
)
axes[0, 1].set_xticks(months_scale)
axes[0, 1].set_xticklabels(month_names, rotation=45, ha="right")
axes[0, 1].set_title('Posterior Mean and +/- 2 standard deviations for monthly SST & SAT')
axes[0, 1].set_xlabel('Month')
axes[0, 1].set_ylabel('Temperature (°C)')
axes[0, 1].set_ylim(0, 30)
axes[0, 1].legend(loc='upper left')
axes[0, 1].grid(True)

# --- SSS & Precipitation Prior and Posterior (Shared Plot, Twin Axes) ---

# Get y-limits for SSS and precipitation (prior and posterior) to keep them consistent
sss_min = min(np.min(mu_prior_SSS_d18Ow_Harwood_monthly_T - 2 * std_prior_SSS_d18Ow_Harwood_monthly_T),
              np.min(mu_post_SSS_d18Ow_monthly_aggregated_Harwood_T - 2 * std_post_SSS_d18Ow_monthly_aggregated_Harwood_T))
sss_max = max(np.max(mu_prior_SSS_d18Ow_Harwood_monthly_T + 2 * std_prior_SSS_d18Ow_Harwood_monthly_T),
              np.max(mu_post_SSS_d18Ow_monthly_aggregated_Harwood_T + 2 * std_post_SSS_d18Ow_monthly_aggregated_Harwood_T))
precip_min = min(np.min(mu_prior_precip_monthly_original - 2 * std_prior_precip_monthly),
                 np.min(mu_post_precip_monthly_aggregated_Harwood - 2 * np.sqrt(np.diag(cov_post_precip_monthly_aggregated_Harwood))))
precip_max = max(np.max(mu_prior_precip_monthly_original + 2 * std_prior_precip_monthly),
                 np.max(mu_post_precip_monthly_aggregated_Harwood + 2 * np.sqrt(np.diag(cov_post_precip_monthly_aggregated_Harwood))))

# --- Prior SSS & Precipitation ---
ax1 = axes[1, 0]
ax2 = ax1.twinx()
lns1 = ax1.plot(months_scale, mu_prior_SSS_d18Ow_Harwood_monthly_T, label='Prior SSS Mean', marker='o', color="green")
fill1 = ax1.fill_between(
    months_scale,
    mu_prior_SSS_d18Ow_Harwood_monthly_T - 2 * std_prior_SSS_d18Ow_Harwood_monthly_T,
    mu_prior_SSS_d18Ow_Harwood_monthly_T + 2 * std_prior_SSS_d18Ow_Harwood_monthly_T,
    color="green", alpha=0.2, label='+/- 2 standard deviations (SSS)'
)
lns2 = ax2.plot(months_scale, mu_prior_precip_monthly_original, label='Prior Precip Mean', marker='o', color="purple")
fill2 = ax2.fill_between(
    months_scale,
    mu_prior_precip_monthly_original - 2 * std_prior_precip_monthly,
    mu_prior_precip_monthly_original + 2 * std_prior_precip_monthly,
    color="purple", alpha=0.2, label='+/- 2 standard deviations (Precip)'
)
ax1.set_xticks(months_scale)
ax1.set_xticklabels(month_names, rotation=45, ha="right")
ax1.set_title('Prior Mean and +/- 2 standard deviations for monthly SSS & Precipitation')
ax1.set_xlabel('Month')
ax1.set_ylabel('Salinity (PSU)', color="green")
ax2.set_ylabel('Precipitation (mm/day)', color="purple")
ax1.set_ylim(sss_min, sss_max)
ax2.set_ylim(precip_min, precip_max)
lns = lns1 + [fill1] + lns2 + [fill2]
labels = [l.get_label() for l in lns1] + [fill1.get_label()] + [l.get_label() for l in lns2] + [fill2.get_label()]
ax1.legend(lns, labels, loc='upper left')
ax1.grid(True)

# --- Posterior SSS & Precipitation ---
ax1 = axes[1, 1]
ax2 = ax1.twinx()
lns1 = ax1.plot(months_scale, mu_post_SSS_d18Ow_monthly_aggregated_Harwood_T, label='Posterior SSS Mean', marker='o', color="green")
fill1 = ax1.fill_between(
    months_scale,
    mu_post_SSS_d18Ow_monthly_aggregated_Harwood_T - 2 * std_post_SSS_d18Ow_monthly_aggregated_Harwood_T,
    mu_post_SSS_d18Ow_monthly_aggregated_Harwood_T + 2 * std_post_SSS_d18Ow_monthly_aggregated_Harwood_T,
    color="green", alpha=0.2, label='+/- 2 standard deviations (SSS)'
)
lns2 = ax2.plot(months_scale, mu_post_precip_monthly_aggregated_Harwood, label='Posterior Precip Mean', marker='o', color="purple")
fill2 = ax2.fill_between(
    months_scale,
    mu_post_precip_monthly_aggregated_Harwood - 2 * np.sqrt(np.diag(cov_post_precip_monthly_aggregated_Harwood)),
    mu_post_precip_monthly_aggregated_Harwood + 2 * np.sqrt(np.diag(cov_post_precip_monthly_aggregated_Harwood)),
    color="purple", alpha=0.2, label='+/- 2 standard deviations (Precip)'
)
ax1.set_xticks(months_scale)
ax1.set_xticklabels(month_names, rotation=45, ha="right")
ax1.set_title('Posterior Mean and +/- 2 standard deviations for monthly SSS & Precipitation')
ax1.set_xlabel('Month')
ax1.set_ylabel('Salinity (PSU)', color="green")
ax2.set_ylabel('Precipitation (mm/day)', color="purple")
ax1.set_ylim(sss_min, sss_max)
ax2.set_ylim(precip_min, precip_max)
lns = lns1 + [fill1] + lns2 + [fill2]
labels = [l.get_label() for l in lns1] + [fill1.get_label()] + [l.get_label() for l in lns2] + [fill2.get_label()]
ax1.legend(lns, labels, loc='upper left')
ax1.grid(True)

# Add "n = x" labels below each x-tick for each subplot to show the number of model values or updates
# Prior SST & SAT
for i, month in enumerate(month_names):
    axes[0, 0].text(
        months_scale[i], axes[0, 0].get_ylim()[0] + 1.5,  # adjust location as needed
        f"n = {n_models_monthly[i] if hasattr(n_models_monthly, '__getitem__') else n_models_monthly}",
        ha='center', va='top', fontsize=10
    )

# Posterior SST & SAT
for i, month in enumerate(month_names):
    axes[0, 1].text(
        months_scale[i], axes[0, 1].get_ylim()[0] + 1.5,
        f"n = {int(n_update_monthly_aggregated_Harwood_D47[i]) if hasattr(n_update_monthly_aggregated_Harwood_D47, '__getitem__') else n_update_monthly_aggregated_Harwood_D47}",
        ha='center', va='top', fontsize=10
    )

# Prior SSS & Precipitation
for i, month in enumerate(month_names):
    axes[1, 0].text(
        months_scale[i], axes[1, 0].get_ylim()[0] + 1.5,
        f"n = {n_models_monthly[i] if hasattr(n_models_monthly, '__getitem__') else n_models_monthly}",
        ha='center', va='top', fontsize=10
    )

# Posterior SSS & Precipitation
for i, month in enumerate(month_names):
    axes[1, 1].text(
        months_scale[i], axes[1, 1].get_ylim()[0] + 1.5,
        f"n = {int(n_update_monthly_aggregated_Harwood_D47[i]) if hasattr(n_update_monthly_aggregated_Harwood_D47, '__getitem__') else n_update_monthly_aggregated_Harwood_D47}",
        ha='center', va='top', fontsize=10
    )

plt.tight_layout()
plt.show()

# Create a figure with two rows and two columns of subplots
fig, axes = plt.subplots(2, 2, figsize=(18, 12))

# --- SST & SAT Prior and Posterior ---
# Prior SST & SAT
axes[0, 0].plot(months_scale, mu_prior_SST_monthly, label='Prior SST Mean', marker='o')
axes[0, 0].plot(months_scale, mu_prior_SAT_monthly, label='Prior SAT Mean', marker='o')
axes[0, 0].fill_between(
    months_scale,
    mu_prior_SST_monthly - 2 * std_prior_SST_monthly,
    mu_prior_SST_monthly + 2 * std_prior_SST_monthly,
    alpha=0.2, label='+/- 2 standard deviations (SST)'
)
axes[0, 0].fill_between(
    months_scale,
    mu_prior_SAT_monthly - 2 * std_prior_SAT_monthly,
    mu_prior_SAT_monthly + 2 * std_prior_SAT_monthly,
    alpha=0.2, label='+/- 2 standard deviations (SAT)'
)
axes[0, 0].set_xticks(months_scale)
axes[0, 0].set_xticklabels(month_names, rotation=45, ha="right")
axes[0, 0].set_title('Prior Mean and +/- 2 standard deviations for monthly SST & SAT')
axes[0, 0].set_xlabel('Month')
axes[0, 0].set_ylabel('Temperature (°C)')
axes[0, 0].set_ylim(0, 30)
axes[0, 0].legend(loc='upper left')
axes[0, 0].grid(True)

# Posterior SST & SAT
axes[0, 1].plot(months_scale, mu_post_SST_D47_monthly_aggregated_Graniero_T, label='Posterior SST Mean', marker='o')
axes[0, 1].plot(months_scale, mu_post_SAT_D47_monthly_aggregated_Graniero_T, label='Posterior SAT Mean', marker='o')
axes[0, 1].fill_between(
    months_scale,
    mu_post_SST_D47_monthly_aggregated_Graniero_T - 2 * std_post_SST_D47_monthly_aggregated_Graniero_T,
    mu_post_SST_D47_monthly_aggregated_Graniero_T + 2 * std_post_SST_D47_monthly_aggregated_Graniero_T,
    alpha=0.2, label='+/- 2 standard deviations (SST)'
)
axes[0, 1].fill_between(
    months_scale,
    mu_post_SAT_D47_monthly_aggregated_Graniero_T - 2 * std_post_SAT_D47_monthly_aggregated_Graniero_T,
    mu_post_SAT_D47_monthly_aggregated_Graniero_T + 2 * std_post_SAT_D47_monthly_aggregated_Graniero_T,
    alpha=0.2, label='+/- 2 standard deviations (SAT)'
)
axes[0, 1].set_xticks(months_scale)
axes[0, 1].set_xticklabels(month_names, rotation=45, ha="right")
axes[0, 1].set_title('Posterior Mean and +/- 2 standard deviations for monthly SST & SAT')
axes[0, 1].set_xlabel('Month')
axes[0, 1].set_ylabel('Temperature (°C)')
axes[0, 1].set_ylim(0, 30)
axes[0, 1].legend(loc='upper left')
axes[0, 1].grid(True)

# --- SSS & Precipitation Prior and Posterior (Shared Plot, Twin Axes) ---

# Get y-limits for SSS and precipitation (prior and posterior) to keep them consistent
sss_min = min(np.min(mu_prior_SSS_d18Ow_Graniero_monthly_T - 2 * std_prior_SSS_d18Ow_Graniero_monthly_T),
              np.min(mu_post_SSS_d18Ow_monthly_aggregated_Graniero_T - 2 * std_post_SSS_d18Ow_monthly_aggregated_Graniero_T))
sss_max = max(np.max(mu_prior_SSS_d18Ow_Graniero_monthly_T + 2 * std_prior_SSS_d18Ow_Graniero_monthly_T),
              np.max(mu_post_SSS_d18Ow_monthly_aggregated_Graniero_T + 2 * std_post_SSS_d18Ow_monthly_aggregated_Graniero_T))
precip_min = min(np.min(mu_prior_precip_monthly_original - 2 * std_prior_precip_monthly),
                 np.min(mu_post_precip_monthly_aggregated_Graniero - 2 * np.sqrt(np.diag(cov_post_precip_monthly_aggregated_Graniero))))
precip_max = max(np.max(mu_prior_precip_monthly_original + 2 * std_prior_precip_monthly),
                 np.max(mu_post_precip_monthly_aggregated_Graniero + 2 * np.sqrt(np.diag(cov_post_precip_monthly_aggregated_Graniero))))

# --- Prior SSS & Precipitation ---
ax1 = axes[1, 0]
ax2 = ax1.twinx()
lns1 = ax1.plot(months_scale, mu_prior_SSS_d18Ow_Graniero_monthly_T, label='Prior SSS Mean', marker='o', color="green")
fill1 = ax1.fill_between(
    months_scale,
    mu_prior_SSS_d18Ow_Graniero_monthly_T - 2 * std_prior_SSS_d18Ow_Graniero_monthly_T,
    mu_prior_SSS_d18Ow_Graniero_monthly_T + 2 * std_prior_SSS_d18Ow_Graniero_monthly_T,
    color="green", alpha=0.2, label='+/- 2 standard deviations (SSS)'
)
lns2 = ax2.plot(months_scale, mu_prior_precip_monthly_original, label='Prior Precip Mean', marker='o', color="purple")
fill2 = ax2.fill_between(
    months_scale,
    mu_prior_precip_monthly_original - 2 * std_prior_precip_monthly,
    mu_prior_precip_monthly_original + 2 * std_prior_precip_monthly,
    color="purple", alpha=0.2, label='+/- 2 standard deviations (Precip)'
)
ax1.set_xticks(months_scale)
ax1.set_xticklabels(month_names, rotation=45, ha="right")
ax1.set_title('Prior Mean and +/- 2 standard deviations for monthly SSS & Precipitation')
ax1.set_xlabel('Month')
ax1.set_ylabel('Salinity (PSU)', color="green")
ax2.set_ylabel('Precipitation (mm/day)', color="purple")
ax1.set_ylim(sss_min, sss_max)
ax2.set_ylim(precip_min, precip_max)
lns = lns1 + [fill1] + lns2 + [fill2]
labels = [l.get_label() for l in lns1] + [fill1.get_label()] + [l.get_label() for l in lns2] + [fill2.get_label()]
ax1.legend(lns, labels, loc='upper left')
ax1.grid(True)

# --- Posterior SSS & Precipitation ---
ax1 = axes[1, 1]
ax2 = ax1.twinx()
lns1 = ax1.plot(months_scale, mu_post_SSS_d18Ow_monthly_aggregated_Graniero_T, label='Posterior SSS Mean', marker='o', color="green")
fill1 = ax1.fill_between(
    months_scale,
    mu_post_SSS_d18Ow_monthly_aggregated_Graniero_T - 2 * std_post_SSS_d18Ow_monthly_aggregated_Graniero_T,
    mu_post_SSS_d18Ow_monthly_aggregated_Graniero_T + 2 * std_post_SSS_d18Ow_monthly_aggregated_Graniero_T,
    color="green", alpha=0.2, label='+/- 2 standard deviations (SSS)'
)
lns2 = ax2.plot(months_scale, mu_post_precip_monthly_aggregated_Graniero, label='Posterior Precip Mean', marker='o', color="purple")
fill2 = ax2.fill_between(
    months_scale,
    mu_post_precip_monthly_aggregated_Graniero - 2 * np.sqrt(np.diag(cov_post_precip_monthly_aggregated_Graniero)),
    mu_post_precip_monthly_aggregated_Graniero + 2 * np.sqrt(np.diag(cov_post_precip_monthly_aggregated_Graniero)),
    color="purple", alpha=0.2, label='+/- 2 standard deviations (Precip)'
)
ax1.set_xticks(months_scale)
ax1.set_xticklabels(month_names, rotation=45, ha="right")
ax1.set_title('Posterior Mean and +/- 2 standard deviations for monthly SSS & Precipitation')
ax1.set_xlabel('Month')
ax1.set_ylabel('Salinity (PSU)', color="green")
ax2.set_ylabel('Precipitation (mm/day)', color="purple")
ax1.set_ylim(sss_min, sss_max)
ax2.set_ylim(precip_min, precip_max)
lns = lns1 + [fill1] + lns2 + [fill2]
labels = [l.get_label() for l in lns1] + [fill1.get_label()] + [l.get_label() for l in lns2] + [fill2.get_label()]
ax1.legend(lns, labels, loc='upper left')
ax1.grid(True)

# Add "n = x" labels below each x-tick for each subplot to show the number of model values or updates
# Prior SST & SAT
for i, month in enumerate(month_names):
    axes[0, 0].text(
        months_scale[i], axes[0, 0].get_ylim()[0] + 1.5,  # adjust location as needed
        f"n = {n_models_monthly[i] if hasattr(n_models_monthly, '__getitem__') else n_models_monthly}",
        ha='center', va='top', fontsize=10
    )

# Posterior SST & SAT
for i, month in enumerate(month_names):
    axes[0, 1].text(
        months_scale[i], axes[0, 1].get_ylim()[0] + 1.5,
        f"n = {int(n_update_monthly_aggregated_Graniero_D47[i]) if hasattr(n_update_monthly_aggregated_Graniero_D47, '__getitem__') else n_update_monthly_aggregated_Graniero_D47}",
        ha='center', va='top', fontsize=10
    )

# Prior SSS & Precipitation
for i, month in enumerate(month_names):
    axes[1, 0].text(
        months_scale[i], axes[1, 0].get_ylim()[0] + 1.5,
        f"n = {n_models_monthly[i] if hasattr(n_models_monthly, '__getitem__') else n_models_monthly}",
        ha='center', va='top', fontsize=10
    )

# Posterior SSS & Precipitation
for i, month in enumerate(month_names):
    axes[1, 1].text(
        months_scale[i], axes[1, 1].get_ylim()[0] + 1.5,
        f"n = {int(n_update_monthly_aggregated_Graniero_D47[i]) if hasattr(n_update_monthly_aggregated_Graniero_D47, '__getitem__') else n_update_monthly_aggregated_Graniero_D47}",
        ha='center', va='top', fontsize=10
    )

plt.tight_layout()
plt.show()

# Create a figure with two rows and two columns of subplots
fig, axes = plt.subplots(2, 2, figsize=(18, 12))

# --- SST & SAT Prior and Posterior ---
# Prior SST & SAT
axes[0, 0].plot(months_scale, mu_prior_SST_monthly, label='Prior SST Mean', marker='o')
axes[0, 0].plot(months_scale, mu_prior_SAT_monthly, label='Prior SAT Mean', marker='o')
axes[0, 0].fill_between(
    months_scale,
    mu_prior_SST_monthly - 2 * std_prior_SST_monthly,
    mu_prior_SST_monthly + 2 * std_prior_SST_monthly,
    alpha=0.2, label='+/- 2 standard deviations (SST)'
)
axes[0, 0].fill_between(
    months_scale,
    mu_prior_SAT_monthly - 2 * std_prior_SAT_monthly,
    mu_prior_SAT_monthly + 2 * std_prior_SAT_monthly,
    alpha=0.2, label='+/- 2 standard deviations (SAT)'
)
axes[0, 0].set_xticks(months_scale)
axes[0, 0].set_xticklabels(month_names, rotation=45, ha="right")
axes[0, 0].set_title('Prior Mean and +/- 2 standard deviations for monthly SST & SAT')
axes[0, 0].set_xlabel('Month')
axes[0, 0].set_ylabel('Temperature (°C)')
axes[0, 0].set_ylim(0, 30)
axes[0, 0].legend(loc='upper left')
axes[0, 0].grid(True)

# Posterior SST & SAT
axes[0, 1].plot(months_scale, mu_post_SST_D47_monthly_aggregated_Railsback_ice_T, label='Posterior SST Mean', marker='o')
axes[0, 1].plot(months_scale, mu_post_SAT_D47_monthly_aggregated_Railsback_ice_T, label='Posterior SAT Mean', marker='o')
axes[0, 1].fill_between(
    months_scale,
    mu_post_SST_D47_monthly_aggregated_Railsback_ice_T - 2 * std_post_SST_D47_monthly_aggregated_Railsback_ice_T,
    mu_post_SST_D47_monthly_aggregated_Railsback_ice_T + 2 * std_post_SST_D47_monthly_aggregated_Railsback_ice_T,
    alpha=0.2, label='+/- 2 standard deviations (SST)'
)
axes[0, 1].fill_between(
    months_scale,
    mu_post_SAT_D47_monthly_aggregated_Railsback_ice_T - 2 * std_post_SAT_D47_monthly_aggregated_Railsback_ice_T,
    mu_post_SAT_D47_monthly_aggregated_Railsback_ice_T + 2 * std_post_SAT_D47_monthly_aggregated_Railsback_ice_T,
    alpha=0.2, label='+/- 2 standard deviations (SAT)'
)
axes[0, 1].set_xticks(months_scale)
axes[0, 1].set_xticklabels(month_names, rotation=45, ha="right")
axes[0, 1].set_title('Posterior Mean and +/- 2 standard deviations for monthly SST & SAT')
axes[0, 1].set_xlabel('Month')
axes[0, 1].set_ylabel('Temperature (°C)')
axes[0, 1].set_ylim(0, 30)
axes[0, 1].legend(loc='upper left')
axes[0, 1].grid(True)

# --- SSS & Precipitation Prior and Posterior (Shared Plot, Twin Axes) ---

# Get y-limits for SSS and precipitation (prior and posterior) to keep them consistent
sss_min = min(np.min(mu_prior_SSS_d18Ow_Railsback_ice_monthly_T - 2 * std_prior_SSS_d18Ow_Railsback_ice_monthly_T),
              np.min(mu_post_SSS_d18Ow_monthly_aggregated_Railsback_ice_T - 2 * std_post_SSS_d18Ow_monthly_aggregated_Railsback_ice_T))
sss_max = max(np.max(mu_prior_SSS_d18Ow_Railsback_ice_monthly_T + 2 * std_prior_SSS_d18Ow_Railsback_ice_monthly_T),
              np.max(mu_post_SSS_d18Ow_monthly_aggregated_Railsback_ice_T + 2 * std_post_SSS_d18Ow_monthly_aggregated_Railsback_ice_T))
precip_min = min(np.min(mu_prior_precip_monthly_original - 2 * std_prior_precip_monthly),
                 np.min(mu_post_precip_monthly_aggregated_Railsback_ice - 2 * np.sqrt(np.diag(cov_post_precip_monthly_aggregated_Railsback_ice))))
precip_max = max(np.max(mu_prior_precip_monthly_original + 2 * std_prior_precip_monthly),
                 np.max(mu_post_precip_monthly_aggregated_Railsback_ice + 2 * np.sqrt(np.diag(cov_post_precip_monthly_aggregated_Railsback_ice))))

# --- Prior SSS & Precipitation ---
ax1 = axes[1, 0]
ax2 = ax1.twinx()
lns1 = ax1.plot(months_scale, mu_prior_SSS_d18Ow_Railsback_ice_monthly_T, label='Prior SSS Mean', marker='o', color="green")
fill1 = ax1.fill_between(
    months_scale,
    mu_prior_SSS_d18Ow_Railsback_ice_monthly_T - 2 * std_prior_SSS_d18Ow_Railsback_ice_monthly_T,
    mu_prior_SSS_d18Ow_Railsback_ice_monthly_T + 2 * std_prior_SSS_d18Ow_Railsback_ice_monthly_T,
    color="green", alpha=0.2, label='+/- 2 standard deviations (SSS)'
)
lns2 = ax2.plot(months_scale, mu_prior_precip_monthly_original, label='Prior Precip Mean', marker='o', color="purple")
fill2 = ax2.fill_between(
    months_scale,
    mu_prior_precip_monthly_original - 2 * std_prior_precip_monthly,
    mu_prior_precip_monthly_original + 2 * std_prior_precip_monthly,
    color="purple", alpha=0.2, label='+/- 2 standard deviations (Precip)'
)
ax1.set_xticks(months_scale)
ax1.set_xticklabels(month_names, rotation=45, ha="right")
ax1.set_title('Prior Mean and +/- 2 standard deviations for monthly SSS & Precipitation')
ax1.set_xlabel('Month')
ax1.set_ylabel('Salinity (PSU)', color="green")
ax2.set_ylabel('Precipitation (mm/day)', color="purple")
ax1.set_ylim(sss_min, sss_max)
ax2.set_ylim(precip_min, precip_max)
lns = lns1 + [fill1] + lns2 + [fill2]
labels = [l.get_label() for l in lns1] + [fill1.get_label()] + [l.get_label() for l in lns2] + [fill2.get_label()]
ax1.legend(lns, labels, loc='upper left')
ax1.grid(True)

# --- Posterior SSS & Precipitation ---
ax1 = axes[1, 1]
ax2 = ax1.twinx()
lns1 = ax1.plot(months_scale, mu_post_SSS_d18Ow_monthly_aggregated_Railsback_ice_T, label='Posterior SSS Mean', marker='o', color="green")
fill1 = ax1.fill_between(
    months_scale,
    mu_post_SSS_d18Ow_monthly_aggregated_Railsback_ice_T - 2 * std_post_SSS_d18Ow_monthly_aggregated_Railsback_ice_T,
    mu_post_SSS_d18Ow_monthly_aggregated_Railsback_ice_T + 2 * std_post_SSS_d18Ow_monthly_aggregated_Railsback_ice_T,
    color="green", alpha=0.2, label='+/- 2 standard deviations (SSS)'
)
lns2 = ax2.plot(months_scale, mu_post_precip_monthly_aggregated_Railsback_ice, label='Posterior Precip Mean', marker='o', color="purple")
fill2 = ax2.fill_between(
    months_scale,
    mu_post_precip_monthly_aggregated_Railsback_ice - 2 * np.sqrt(np.diag(cov_post_precip_monthly_aggregated_Railsback_ice)),
    mu_post_precip_monthly_aggregated_Railsback_ice + 2 * np.sqrt(np.diag(cov_post_precip_monthly_aggregated_Railsback_ice)),
    color="purple", alpha=0.2, label='+/- 2 standard deviations (Precip)'
)
ax1.set_xticks(months_scale)
ax1.set_xticklabels(month_names, rotation=45, ha="right")
ax1.set_title('Posterior Mean and +/- 2 standard deviations for monthly SSS & Precipitation')
ax1.set_xlabel('Month')
ax1.set_ylabel('Salinity (PSU)', color="green")
ax2.set_ylabel('Precipitation (mm/day)', color="purple")
ax1.set_ylim(sss_min, sss_max)
ax2.set_ylim(precip_min, precip_max)
lns = lns1 + [fill1] + lns2 + [fill2]
labels = [l.get_label() for l in lns1] + [fill1.get_label()] + [l.get_label() for l in lns2] + [fill2.get_label()]
ax1.legend(lns, labels, loc='upper left')
ax1.grid(True)

# Add "n = x" labels below each x-tick for each subplot to show the number of model values or updates
# Prior SST & SAT
for i, month in enumerate(month_names):
    axes[0, 0].text(
        months_scale[i], axes[0, 0].get_ylim()[0] + 1.5,  # adjust location as needed
        f"n = {n_models_monthly[i] if hasattr(n_models_monthly, '__getitem__') else n_models_monthly}",
        ha='center', va='top', fontsize=10
    )

# Posterior SST & SAT
for i, month in enumerate(month_names):
    axes[0, 1].text(
        months_scale[i], axes[0, 1].get_ylim()[0] + 1.5,
        f"n = {int(n_update_monthly_aggregated_Railsback_ice_D47[i]) if hasattr(n_update_monthly_aggregated_Railsback_ice_D47, '__getitem__') else n_update_monthly_aggregated_Railsback_ice_D47}",
        ha='center', va='top', fontsize=10
    )

# Prior SSS & Precipitation
for i, month in enumerate(month_names):
    axes[1, 0].text(
        months_scale[i], axes[1, 0].get_ylim()[0] + 1.5,
        f"n = {n_models_monthly[i] if hasattr(n_models_monthly, '__getitem__') else n_models_monthly}",
        ha='center', va='top', fontsize=10
    )

# Posterior SSS & Precipitation
for i, month in enumerate(month_names):
    axes[1, 1].text(
        months_scale[i], axes[1, 1].get_ylim()[0] + 1.5,
        f"n = {int(n_update_monthly_aggregated_Railsback_ice_D47[i]) if hasattr(n_update_monthly_aggregated_Railsback_ice_D47, '__getitem__') else n_update_monthly_aggregated_Railsback_ice_D47}",
        ha='center', va='top', fontsize=10
    )

plt.tight_layout()
plt.show()

# Create a figure with two rows and two columns of subplots
fig, axes = plt.subplots(2, 2, figsize=(18, 12))

# --- SST & SAT Prior and Posterior ---
# Prior SST & SAT
axes[0, 0].plot(months_scale, mu_prior_SST_monthly, label='Prior SST Mean', marker='o')
axes[0, 0].plot(months_scale, mu_prior_SAT_monthly, label='Prior SAT Mean', marker='o')
axes[0, 0].fill_between(
    months_scale,
    mu_prior_SST_monthly - 2 * std_prior_SST_monthly,
    mu_prior_SST_monthly + 2 * std_prior_SST_monthly,
    alpha=0.2, label='+/- 2 standard deviations (SST)'
)
axes[0, 0].fill_between(
    months_scale,
    mu_prior_SAT_monthly - 2 * std_prior_SAT_monthly,
    mu_prior_SAT_monthly + 2 * std_prior_SAT_monthly,
    alpha=0.2, label='+/- 2 standard deviations (SAT)'
)
axes[0, 0].set_xticks(months_scale)
axes[0, 0].set_xticklabels(month_names, rotation=45, ha="right")
axes[0, 0].set_title('Prior Mean and +/- 2 standard deviations for monthly SST & SAT')
axes[0, 0].set_xlabel('Month')
axes[0, 0].set_ylabel('Temperature (°C)')
axes[0, 0].set_ylim(0, 30)
axes[0, 0].legend(loc='upper left')
axes[0, 0].grid(True)

# Posterior SST & SAT
axes[0, 1].plot(months_scale, mu_post_SST_D47_monthly_aggregated_Railsback_green_T, label='Posterior SST Mean', marker='o')
axes[0, 1].plot(months_scale, mu_post_SAT_D47_monthly_aggregated_Railsback_green_T, label='Posterior SAT Mean', marker='o')
axes[0, 1].fill_between(
    months_scale,
    mu_post_SST_D47_monthly_aggregated_Railsback_green_T - 2 * std_post_SST_D47_monthly_aggregated_Railsback_green_T,
    mu_post_SST_D47_monthly_aggregated_Railsback_green_T + 2 * std_post_SST_D47_monthly_aggregated_Railsback_green_T,
    alpha=0.2, label='+/- 2 standard deviations (SST)'
)
axes[0, 1].fill_between(
    months_scale,
    mu_post_SAT_D47_monthly_aggregated_Railsback_green_T - 2 * std_post_SAT_D47_monthly_aggregated_Railsback_green_T,
    mu_post_SAT_D47_monthly_aggregated_Railsback_green_T + 2 * std_post_SAT_D47_monthly_aggregated_Railsback_green_T,
    alpha=0.2, label='+/- 2 standard deviations (SAT)'
)
axes[0, 1].set_xticks(months_scale)
axes[0, 1].set_xticklabels(month_names, rotation=45, ha="right")
axes[0, 1].set_title('Posterior Mean and +/- 2 standard deviations for monthly SST & SAT')
axes[0, 1].set_xlabel('Month')
axes[0, 1].set_ylabel('Temperature (°C)')
axes[0, 1].set_ylim(0, 30)
axes[0, 1].legend(loc='upper left')
axes[0, 1].grid(True)

# --- SSS & Precipitation Prior and Posterior (Shared Plot, Twin Axes) ---

# Get y-limits for SSS and precipitation (prior and posterior) to keep them consistent
sss_min = min(np.min(mu_prior_SSS_d18Ow_Railsback_green_monthly_T - 2 * std_prior_SSS_d18Ow_Railsback_green_monthly_T),
              np.min(mu_post_SSS_d18Ow_monthly_aggregated_Railsback_green_T - 2 * std_post_SSS_d18Ow_monthly_aggregated_Railsback_green_T))
sss_max = max(np.max(mu_prior_SSS_d18Ow_Railsback_green_monthly_T + 2 * std_prior_SSS_d18Ow_Railsback_green_monthly_T),
              np.max(mu_post_SSS_d18Ow_monthly_aggregated_Railsback_green_T + 2 * std_post_SSS_d18Ow_monthly_aggregated_Railsback_green_T))
precip_min = min(np.min(mu_prior_precip_monthly_original - 2 * std_prior_precip_monthly),
                 np.min(mu_post_precip_monthly_aggregated_Railsback_green - 2 * np.sqrt(np.diag(cov_post_precip_monthly_aggregated_Railsback_green))))
precip_max = max(np.max(mu_prior_precip_monthly_original + 2 * std_prior_precip_monthly),
                 np.max(mu_post_precip_monthly_aggregated_Railsback_green + 2 * np.sqrt(np.diag(cov_post_precip_monthly_aggregated_Railsback_green))))

# --- Prior SSS & Precipitation ---
ax1 = axes[1, 0]
ax2 = ax1.twinx()
lns1 = ax1.plot(months_scale, mu_prior_SSS_d18Ow_Railsback_green_monthly_T, label='Prior SSS Mean', marker='o', color="green")
fill1 = ax1.fill_between(
    months_scale,
    mu_prior_SSS_d18Ow_Railsback_green_monthly_T - 2 * std_prior_SSS_d18Ow_Railsback_green_monthly_T,
    mu_prior_SSS_d18Ow_Railsback_green_monthly_T + 2 * std_prior_SSS_d18Ow_Railsback_green_monthly_T,
    color="green", alpha=0.2, label='+/- 2 standard deviations (SSS)'
)
lns2 = ax2.plot(months_scale, mu_prior_precip_monthly_original, label='Prior Precip Mean', marker='o', color="purple")
fill2 = ax2.fill_between(
    months_scale,
    mu_prior_precip_monthly_original - 2 * std_prior_precip_monthly,
    mu_prior_precip_monthly_original + 2 * std_prior_precip_monthly,
    color="purple", alpha=0.2, label='+/- 2 standard deviations (Precip)'
)
ax1.set_xticks(months_scale)
ax1.set_xticklabels(month_names, rotation=45, ha="right")
ax1.set_title('Prior Mean and +/- 2 standard deviations for monthly SSS & Precipitation')
ax1.set_xlabel('Month')
ax1.set_ylabel('Salinity (PSU)', color="green")
ax2.set_ylabel('Precipitation (mm/day)', color="purple")
ax1.set_ylim(sss_min, sss_max)
ax2.set_ylim(precip_min, precip_max)
lns = lns1 + [fill1] + lns2 + [fill2]
labels = [l.get_label() for l in lns1] + [fill1.get_label()] + [l.get_label() for l in lns2] + [fill2.get_label()]
ax1.legend(lns, labels, loc='upper left')
ax1.grid(True)

# --- Posterior SSS & Precipitation ---
ax1 = axes[1, 1]
ax2 = ax1.twinx()
lns1 = ax1.plot(months_scale, mu_post_SSS_d18Ow_monthly_aggregated_Railsback_green_T, label='Posterior SSS Mean', marker='o', color="green")
fill1 = ax1.fill_between(
    months_scale,
    mu_post_SSS_d18Ow_monthly_aggregated_Railsback_green_T - 2 * std_post_SSS_d18Ow_monthly_aggregated_Railsback_green_T,
    mu_post_SSS_d18Ow_monthly_aggregated_Railsback_green_T + 2 * std_post_SSS_d18Ow_monthly_aggregated_Railsback_green_T,
    color="green", alpha=0.2, label='+/- 2 standard deviations (SSS)'
)
lns2 = ax2.plot(months_scale, mu_post_precip_monthly_aggregated_Railsback_green, label='Posterior Precip Mean', marker='o', color="purple")
fill2 = ax2.fill_between(
    months_scale,
    mu_post_precip_monthly_aggregated_Railsback_green - 2 * np.sqrt(np.diag(cov_post_precip_monthly_aggregated_Railsback_green)),
    mu_post_precip_monthly_aggregated_Railsback_green + 2 * np.sqrt(np.diag(cov_post_precip_monthly_aggregated_Railsback_green)),
    color="purple", alpha=0.2, label='+/- 2 standard deviations (Precip)'
)
ax1.set_xticks(months_scale)
ax1.set_xticklabels(month_names, rotation=45, ha="right")
ax1.set_title('Posterior Mean and +/- 2 standard deviations for monthly SSS & Precipitation')
ax1.set_xlabel('Month')
ax1.set_ylabel('Salinity (PSU)', color="green")
ax2.set_ylabel('Precipitation (mm/day)', color="purple")
ax1.set_ylim(sss_min, sss_max)
ax2.set_ylim(precip_min, precip_max)
lns = lns1 + [fill1] + lns2 + [fill2]
labels = [l.get_label() for l in lns1] + [fill1.get_label()] + [l.get_label() for l in lns2] + [fill2.get_label()]
ax1.legend(lns, labels, loc='upper left')
ax1.grid(True)

# Add "n = x" labels below each x-tick for each subplot to show the number of model values or updates
# Prior SST & SAT
for i, month in enumerate(month_names):
    axes[0, 0].text(
        months_scale[i], axes[0, 0].get_ylim()[0] + 1.5,  # adjust location as needed
        f"n = {n_models_monthly[i] if hasattr(n_models_monthly, '__getitem__') else n_models_monthly}",
        ha='center', va='top', fontsize=10
    )

# Posterior SST & SAT
for i, month in enumerate(month_names):
    axes[0, 1].text(
        months_scale[i], axes[0, 1].get_ylim()[0] + 1.5,
        f"n = {int(n_update_monthly_aggregated_Railsback_green_D47[i]) if hasattr(n_update_monthly_aggregated_Railsback_green_D47, '__getitem__') else n_update_monthly_aggregated_Railsback_green_D47}",
        ha='center', va='top', fontsize=10
    )

# Prior SSS & Precipitation
for i, month in enumerate(month_names):
    axes[1, 0].text(
        months_scale[i], axes[1, 0].get_ylim()[0] + 1.5,
        f"n = {n_models_monthly[i] if hasattr(n_models_monthly, '__getitem__') else n_models_monthly}",
        ha='center', va='top', fontsize=10
    )

# Posterior SSS & Precipitation
for i, month in enumerate(month_names):
    axes[1, 1].text(
        months_scale[i], axes[1, 1].get_ylim()[0] + 1.5,
        f"n = {int(n_update_monthly_aggregated_Railsback_green_D47[i]) if hasattr(n_update_monthly_aggregated_Railsback_green_D47, '__getitem__') else n_update_monthly_aggregated_Railsback_green_D47}",
        ha='center', va='top', fontsize=10
    )

plt.tight_layout()
plt.show()

# Aggregate SST outcomes and their precisions (standard deviations) based on assimilations with different d18Ow-SSS relationships into a table

# Create a dictionary to store the results
results = {
    'Month': month_names,
    'Harwood_Mean': mu_post_SST_D47_monthly_aggregated_Harwood_T,
    'Harwood_SD': std_post_SST_D47_monthly_aggregated_Harwood_T,
    'Graniero_Mean': mu_post_SST_D47_monthly_aggregated_Graniero_T,
    'Graniero_SD': std_post_SST_D47_monthly_aggregated_Graniero_T,
    'Railsback_ice_Mean': mu_post_SST_D47_monthly_aggregated_Railsback_ice_T,
    'Railsback_ice_SD': std_post_SST_D47_monthly_aggregated_Railsback_ice_T,
    'Railsback_green_Mean': mu_post_SST_D47_monthly_aggregated_Railsback_green_T,
    'Railsback_green_SD': std_post_SST_D47_monthly_aggregated_Railsback_green_T
}

# Create a Pandas DataFrame from the dictionary
df_results = pd.DataFrame(results)

# Export the table to a CSV file
df_results.to_csv("Sensitivity_test_d18Ow_SSS_relationship/posterior_SST_outcomes_table.csv")

# Print the DataFrame
print(df_results)

        Month  Harwood_Mean  Harwood_SD  Graniero_Mean  Graniero_SD  \
0     January      8.530402    0.926335       8.528514     0.933579   
1    February      6.536536    0.953439       6.528670     0.959274   
2       March      5.860544    0.964259       5.840724     0.960354   
3       April      7.591868    0.941116       7.578535     0.940751   
4         May     10.532464    1.001044      10.515754     1.003307   
5        June     13.338931    1.342293      13.321306     1.344975   
6        July     16.133690    1.629708      16.122995     1.634835   
7      August     17.354045    1.697047      17.365756     1.710862   
8   September     17.992636    1.541160      18.007466     1.565430   
9     October     16.642851    1.524642      16.660066     1.553228   
10   November     13.952476    1.362536      13.970987     1.388602   
11   December     12.041142    1.003713      12.052362     1.019625   

    Railsback_ice_Mean  Railsback_ice_SD  Railsback_green_Mean  \
0             8.570818          0.900527              8.759055   
1             6.535183          0.925796              6.652095   
2             5.754855          0.958412              5.620753   
3             7.509526          0.925532              7.473914   
4            10.500667          0.974664             10.479739   
5            13.249424          1.301558             13.215710   
6            16.010968          1.577663             16.067708   
7            17.171286          1.607201             17.503742   
8            17.970441          1.436618             18.500357   
9            16.739674          1.419482             17.371620   
10           14.076032          1.266400             14.679276   
11           12.202340          0.951116             12.637394   

    Railsback_green_SD  
0             0.935767  
1             0.960812  
2             0.958314  
3             0.940132  
4             1.003111  
5             1.343785  
6             1.634122  
7             1.713568  
8             1.573917  
9             1.565363  
10            1.399186  
11            1.025653

# Aggregate SST innovations and their precisions (standard deviations) based on assimilations with different d18Ow-SSS relationships into a table

# Create a dictionary to store the results
results_innov = {
    'Month': month_names,
    'Harwood_innovation': mu_post_SST_D47_monthly_aggregated_Harwood_T - mu_prior_SST_monthly,
    'Harwood_SD': std_post_SST_D47_monthly_aggregated_Harwood_T,
    'Graniero_innovation': mu_post_SST_D47_monthly_aggregated_Graniero_T - mu_prior_SST_monthly,
    'Graniero_SD': std_post_SST_D47_monthly_aggregated_Graniero_T,
    'Railsback_ice_innovation': mu_post_SST_D47_monthly_aggregated_Railsback_ice_T - mu_prior_SST_monthly,
    'Railsback_ice_SD': std_post_SST_D47_monthly_aggregated_Railsback_ice_T,
    'Railsback_green_innovation': mu_post_SST_D47_monthly_aggregated_Railsback_green_T - mu_prior_SST_monthly,
    'Railsback_green_SD': std_post_SST_D47_monthly_aggregated_Railsback_green_T
}

# Create a Pandas DataFrame from the dictionary
df_results_innov = pd.DataFrame(results_innov)

# Export the table to a CSV file
df_results_innov.to_csv("Sensitivity_test_d18Ow_SSS_relationship/innovation_SST_outcomes_table.csv")

# Print the DataFrame
print(df_results_innov)

        Month  Harwood_innovation  Harwood_SD  Graniero_innovation  \
0     January           -0.368689    0.926335            -0.370577   
1    February           -1.305282    0.953439            -1.313148   
2       March           -2.031274    0.964259            -2.051094   
3       April           -2.017223    0.941116            -2.030556   
4         May           -2.440263    1.001044            -2.456973   
5        June           -4.040160    1.342293            -4.057785   
6        July           -4.966310    1.629708            -4.977005   
7      August           -4.793228    1.697047            -4.781517   
8   September           -2.393727    1.541160            -2.378897   
9     October           -0.609876    1.524642            -0.592661   
10   November            0.120658    1.362536             0.139169   
11   December            1.122961    1.003713             1.134180   

    Graniero_SD  Railsback_ice_innovation  Railsback_ice_SD  \
0      0.933579                 -0.328273          0.900527   
1      0.959274                 -1.306635          0.925796   
2      0.960354                 -2.136964          0.958412   
3      0.940751                 -2.099565          0.925532   
4      1.003307                 -2.472060          0.974664   
5      1.344975                 -4.129667          1.301558   
6      1.634835                 -5.089032          1.577663   
7      1.710862                 -4.975986          1.607201   
8      1.565430                 -2.415922          1.436618   
9      1.553228                 -0.513054          1.419482   
10     1.388602                  0.244214          1.266400   
11     1.019625                  1.284158          0.951116   

    Railsback_green_innovation  Railsback_green_SD  
0                    -0.140036            0.935767  
1                    -1.189724            0.960812  
2                    -2.271065            0.958314  
3                    -2.135177            0.940132  
4                    -2.492988            1.003111  
5                    -4.163380            1.343785  
6                    -5.032292            1.634122  
7                    -4.643531            1.713568  
8                    -1.886007            1.573917  
9                     0.118893            1.565363  
10                    0.847458            1.399186  
11                    1.719213            1.025653

# Aggregate SST innovations and their precisions (standard deviations) based on assimilations with different d18Ow-SSS relationships into a table

# Create a dictionary to store the results
results_SST = {
    'Month': month_names,
    'Prior_mu_SST': mu_prior_SST_monthly,
    'Prior_std_SST': std_prior_SST_monthly,
    'Harwood_mu_SST': mu_post_SST_D47_monthly_aggregated_Harwood_T,
    'Harwood_std_SST': std_post_SST_D47_monthly_aggregated_Harwood_T,
    'Graniero_mu_SST': mu_post_SST_D47_monthly_aggregated_Graniero_T,
    'Graniero_std_SST': std_post_SST_D47_monthly_aggregated_Graniero_T,
    'Railsback_ice_mu_SST': mu_post_SST_D47_monthly_aggregated_Railsback_ice_T,
    'Railsback_ice_std_SST': std_post_SST_D47_monthly_aggregated_Railsback_ice_T,
    'Railsback_green_mu_SST': mu_post_SST_D47_monthly_aggregated_Railsback_green_T,
    'Railsback_green_std_SST': std_post_SST_D47_monthly_aggregated_Railsback_green_T
}

results_SAT = {
    'Month': month_names,
    'Prior_mu_SAT': mu_prior_SAT_monthly,
    'Prior_std_SAT': std_prior_SAT_monthly,
    'Harwood_mu_SAT': mu_post_SAT_D47_monthly_aggregated_Harwood_T,
    'Harwood_std_SAT': std_post_SAT_D47_monthly_aggregated_Harwood_T,
    'Graniero_mu_SAT': mu_post_SAT_D47_monthly_aggregated_Graniero_T,
    'Graniero_std_SAT': std_post_SAT_D47_monthly_aggregated_Graniero_T,
    'Railsback_ice_mu_SAT': mu_post_SAT_D47_monthly_aggregated_Railsback_ice_T,
    'Railsback_ice_std_SAT': std_post_SAT_D47_monthly_aggregated_Railsback_ice_T,
    'Railsback_green_mu_SAT': mu_post_SAT_D47_monthly_aggregated_Railsback_green_T,
    'Railsback_green_std_SAT': std_post_SAT_D47_monthly_aggregated_Railsback_green_T
}

# Create a Pandas DataFrame from the dictionary
df_results_SST = pd.DataFrame(results_SST)
df_results_SAT = pd.DataFrame(results_SAT)

# Export the table to a CSV file
df_results_SST.to_csv("Sensitivity_test_d18Ow_SSS_relationship/SST_outcomes_SSS-d18Ow.csv")
df_results_SAT.to_csv("Sensitivity_test_d18Ow_SSS_relationship/SAT_outcomes_SSS-d18Ow.csv")

# Print the DataFrame
print(df_results_SST)

        Month  Prior_mu_SST  Prior_std_SST  Harwood_mu_SST  Harwood_std_SST  \
0     January      8.899091       1.665193        8.530402         0.926335   
1    February      7.841818       1.705631        6.536536         0.953439   
2       March      7.891818       1.629790        5.860544         0.964259   
3       April      9.609091       1.419440        7.591868         0.941116   
4         May     12.972727       1.740265       10.532464         1.001044   
5        June     17.379091       2.500278       13.338931         1.342293   
6        July     21.100000       3.072107       16.133690         1.629708   
7      August     22.147273       3.321256       17.354045         1.697047   
8   September     20.386364       3.310049       17.992636         1.541160   
9     October     17.252727       2.947457       16.642851         1.524642   
10   November     13.831818       2.348088       13.952476         1.362536   
11   December     10.918182       1.864939       12.041142         1.003713   

    Graniero_mu_SST  Graniero_std_SST  Railsback_ice_mu_SST  \
0          8.528514          0.933579              8.570818   
1          6.528670          0.959274              6.535183   
2          5.840724          0.960354              5.754855   
3          7.578535          0.940751              7.509526   
4         10.515754          1.003307             10.500667   
5         13.321306          1.344975             13.249424   
6         16.122995          1.634835             16.010968   
7         17.365756          1.710862             17.171286   
8         18.007466          1.565430             17.970441   
9         16.660066          1.553228             16.739674   
10        13.970987          1.388602             14.076032   
11        12.052362          1.019625             12.202340   

    Railsback_ice_std_SST  Railsback_green_mu_SST  Railsback_green_std_SST  
0                0.900527                8.759055                 0.935767  
1                0.925796                6.652095                 0.960812  
2                0.958412                5.620753                 0.958314  
3                0.925532                7.473914                 0.940132  
4                0.974664               10.479739                 1.003111  
5                1.301558               13.215710                 1.343785  
6                1.577663               16.067708                 1.634122  
7                1.607201               17.503742                 1.713568  
8                1.436618               18.500357                 1.573917  
9                1.419482               17.371620                 1.565363  
10               1.266400               14.679276                 1.399186  
11               0.951116               12.637394                 1.025653

# Set up the plot
fig, ((ax1, ax2)) = plt.subplots(
    1, 2, figsize=(15, 6), sharex=True
)
fig.subplots_adjust(hspace=0.3, wspace=0.25)  # reduce vertical & horizontal spacing

# Define the SSS-d18Ow relationships
relationships = ['Harwood et al., 2008\n(North Sea)', 'Graniero et al., 2017\n(Panama)', 'Railsback et al., 1989\n(icehouse conditions)', 'Railsback et al., 1989\n(greenhouse conditions)']

# Define the months and their corresponding colors
months_list = df_results_innov['Month'].tolist()
num_months = len(months_list)
cmap = plt.cm.coolwarm
colors = [cmap(i / (len(month_names)-1)) for i in range(len(month_names))]
color_dict = dict(zip(month_names, colors))

# Plot each month as a point for each SSS-d18Ow relationship
for i, month in enumerate(months_list):
    x = relationships  # X-axis is the SSS-d18Ow relationship
    y = [df_results_innov[df_results_innov['Month'] == month]['Harwood_innovation'].values[0],
         df_results_innov[df_results_innov['Month'] == month]['Graniero_innovation'].values[0],
         df_results_innov[df_results_innov['Month'] == month]['Railsback_ice_innovation'].values[0],
         df_results_innov[df_results_innov['Month'] == month]['Railsback_green_innovation'].values[0]]
    yerr = [df_results_innov[df_results_innov['Month'] == month]['Harwood_SD'].values[0],
            df_results_innov[df_results_innov['Month'] == month]['Graniero_SD'].values[0],
            df_results_innov[df_results_innov['Month'] == month]['Railsback_ice_SD'].values[0],
            df_results_innov[df_results_innov['Month'] == month]['Railsback_green_SD'].values[0]]
    
    ax1.scatter(x, y, label=month, marker='o', color=color_dict[month])
    ax2.scatter(x, yerr, label=month, marker='o', color=color_dict[month])

# Customize the plot
ax1.set_xlabel('SSS-d18Ow Relationship')
ax1.set_ylabel('SST Innovation (°C)')
ax1.set_xticks(x)
ax1.set_xticklabels(x, rotation=45, ha='right')
ax1.grid(True)

ax2.set_xlabel('SSS-d18Ow Relationship')
ax2.set_ylabel('SST Uncertainty (°C)')
ax2.set_xticks(x)
ax2.set_xticklabels(x, rotation=45, ha='right')
ax2.grid(True)
plt.tight_layout()

# ---------------------------------------------------------
# Shared legend above all plots
handles1, labels1 = ax1.get_legend_handles_labels()
fig.legend(handles1, labels1,
           loc="upper center",
           bbox_to_anchor=(0.5, 1.05),
           ncol=len(month_names),
           frameon=False)

plt.show()

# Set up the plot
fig, ((ax1, ax2)) = plt.subplots(
    1, 2, figsize=(15, 6), sharex=True
)
fig.subplots_adjust(hspace=0.3, wspace=0.25)  # reduce vertical & horizontal spacing

# Define the x-axis labels based on SSS-d18Ow cases and prior
cases = ['Prior', 'Harwood et al., 2008\n(North Sea)', 'Graniero et al., 2017\n(Panama)', 'Railsback et al., 1989\n(icehouse conditions)', 'Railsback et al., 1989\n(greenhouse conditions)']

# Define the months and their corresponding colors
months_list = df_results_SST['Month'].tolist()
num_months = len(months_list)
cmap = plt.cm.coolwarm
colors = [cmap(i / (len(month_names)-1)) for i in range(len(month_names))]
color_dict = dict(zip(month_names, colors))

# Plot each month as a point for each SSS-d18Ow relationship
for i, month in enumerate(months_list):
    x = cases  # X-axis is the SSS-d18Ow relationship
    y = [df_results_SST[df_results_SST['Month'] == month]['Prior_mu_SST'].values[0],
        df_results_SST[df_results_SST['Month'] == month]['Harwood_mu_SST'].values[0],
        df_results_SST[df_results_SST['Month'] == month]['Graniero_mu_SST'].values[0],
        df_results_SST[df_results_SST['Month'] == month]['Railsback_ice_mu_SST'].values[0],
        df_results_SST[df_results_SST['Month'] == month]['Railsback_green_mu_SST'].values[0]]
    yerr = [df_results_SST[df_results_SST['Month'] == month]['Prior_std_SST'].values[0],
        df_results_SST[df_results_SST['Month'] == month]['Harwood_std_SST'].values[0],
        df_results_SST[df_results_SST['Month'] == month]['Graniero_std_SST'].values[0],
        df_results_SST[df_results_SST['Month'] == month]['Railsback_ice_std_SST'].values[0],
        df_results_SST[df_results_SST['Month'] == month]['Railsback_green_std_SST'].values[0]]
    
    ax1.errorbar(x, y, yerr, label=month, fmt='o', color=color_dict[month])
    # ax1.scatter(x, y, label=month, marker='o', color=color_dict[month])
    # ax2.scatter(x, yerr, label=month, marker='o', color=color_dict[month])

# Customize the plot
ax1.set_xlabel('SSS-d18Ow Relationship')
ax1.set_ylabel('SST Innovation (°C)')
ax1.set_xticks(x)
ax1.set_xticklabels(x, rotation=45, ha='right')
ax1.grid(True)

ax2.set_xlabel('SSS-d18Ow Relationship')
ax2.set_ylabel('SST Uncertainty (°C)')
ax2.set_xticks(x)
ax2.set_xticklabels(x, rotation=45, ha='right')
ax2.grid(True)
plt.tight_layout()

# ---------------------------------------------------------
# Shared legend above all plots
handles1, labels1 = ax1.get_legend_handles_labels()
fig.legend(handles1, labels1,
           loc="upper center",
           bbox_to_anchor=(0.5, 1.05),
           ncol=len(month_names),
           frameon=False)

plt.show()

fig, (ax1, ax3) = plt.subplots(1, 2, figsize=(10, 5), sharey=True)
fig.subplots_adjust(hspace=0.3)

# Columns to plot (mu + corresponding std) and nicer xlabels
mu_cols_sst = ["Prior_mu_SST", "Harwood_mu_SST", "Graniero_mu_SST", "Railsback_ice_mu_SST", "Railsback_green_mu_SST"]
std_cols_sst = ["Prior_std_SST", "Harwood_std_SST", "Graniero_std_SST", "Railsback_ice_std_SST", "Railsback_green_std_SST"]
mu_cols_sat = ["Prior_mu_SAT", "Harwood_mu_SAT", "Graniero_mu_SAT", "Railsback_ice_mu_SAT", "Railsback_green_mu_SAT"]
std_cols_sat = ["Prior_std_SAT", "Harwood_std_SAT", "Graniero_std_SAT", "Railsback_ice_std_SAT", "Railsback_green_std_SAT"]

# Horizontal axis layout prep
x = np.arange(len(mu_cols_sst))
x_offsets = np.linspace(-0.3, 0.3, len(df_results_SST.index))
xlabels = ['Prior', 'Harwood et al., 2008\n(North Sea)', 'Graniero et al., 2017\n(Panama)', 'Railsback et al., 1989\n(icehouse conditions)', 'Railsback et al., 1989\n(greenhouse conditions)']

# colour map for test cases (rows)
# df_results_SST["Month"] = ["January", "February", "March", "April", "May", "June", "July", "August", "September", "October", "November", "December"]
# n_cases = len(df_results_SST)
cmap = plt.cm.get_cmap("coolwarm")
# Check if length of cases equals the number of df_results_SST["Month"] (it should)
colors = [cmap(i / (len(df_results_SST["Month"]) - 1)) for i in range(len(df_results_SST["Month"]))]
legend_labels = df_results_SST["Month"].copy()
# else:
#     colors = [cmap(i / max(n_cases - 1, 1)) for i in range(n_cases)]
#     legend_labels = [str(idx) for idx in df_results_SST.index]

# Plot SST outcomes
for i, idx in enumerate(df_results_SST.index):
    means = df_results_SST.loc[idx, mu_cols_sst].values.astype(float)
    errs = df_results_SST.loc[idx, std_cols_sst].values.astype(float)
    ax1.errorbar(x + x_offsets[i], means, yerr=errs, capsize=3, fmt='o', color=colors[i], alpha=0.9)
ax1.set_ylim(0, 30) # Equalize vertical axes between plot panels
# Add light green rectangle to highlight prior
ax1.add_patch(
    Rectangle(
        (-0.5, ax1.get_ylim()[0]),  # (x,y)
        1,                          # width
        ax1.get_ylim()[1] - ax1.get_ylim()[0],  # height
        color='lightgreen',
        alpha=0.3,
    )
)
# Add light red rectangle to highlight posteriors
ax1.add_patch(
    Rectangle(
        (0.5, ax1.get_ylim()[0]),  # (x,y)
        4,         # width
        ax1.get_ylim()[1] - ax1.get_ylim()[0],  # height
        color='red',
        alpha=0.1,
    )
)
ax1.set_xticks(x)
ax1.set_xticklabels(xlabels, rotation = 30, ha = "right")
ax1.set_ylabel("SST (°C)")

# Plot SAT outcomes
for i, idx in enumerate(df_results_SAT.index):
    means = df_results_SAT.loc[idx, mu_cols_sat].values.astype(float)
    errs = df_results_SAT.loc[idx, std_cols_sat].values.astype(float)
    ax3.errorbar(x + x_offsets[i], means, yerr=errs, capsize=3, fmt='o', color=colors[i], alpha=0.9)
ax3.set_ylim(0, 30) # Equalize vertical axes between plot panels
# Add light green rectangle to highlight prior
ax3.add_patch(
    Rectangle(
        (-0.5, ax3.get_ylim()[0]),  # (x,y)
        1,                          # width
        ax3.get_ylim()[1] - ax3.get_ylim()[0],  # height
        color='lightgreen',
        alpha=0.3,
    )
)
# Add light red rectangle to highlight posteriors
ax3.add_patch(
    Rectangle(
        (0.5, ax3.get_ylim()[0]),  # (x,y)
        4,         # width
        ax3.get_ylim()[1] - ax3.get_ylim()[0],  # height
        color='red',
        alpha=0.1,
    )
)
ax3.set_xticks(x)
ax3.set_xticklabels(xlabels, rotation = 30, ha = "right")
ax3.set_ylabel("SAT (°C)")

# build shared legend (ordered according to legend_labels)
proxy_handles = [Line2D([0], [0], marker='o', color='w', markerfacecolor=colors[i], markersize=8, linestyle='') 
                 for i in range(len(legend_labels))]
fig.legend(proxy_handles, legend_labels, loc="upper center", bbox_to_anchor=(0.5, 1.01), ncol=6, frameon=False)

plt.tight_layout(rect=[0, 0, 1, 0.9])
plt.show()

C:\Users\nwi213\AppData\Local\Temp\ipykernel_19856\2990353717.py:18: MatplotlibDeprecationWarning: The get_cmap function was deprecated in Matplotlib 3.7 and will be removed two minor releases later. Use ``matplotlib.colormaps[name]`` or ``matplotlib.colormaps.get_cmap(obj)`` instead.
  cmap = plt.cm.get_cmap("coolwarm")

	modelname	ja	fb	mr	ar	my	jn	jl	ag	sp	ot	nv	dc
0	CCSM4-UoT	8.33	7.14	7.45	9.67	13.33	18.40	22.50	24.35	22.60	18.74	14.83	10.86
1	CCSM4-Utr	9.96	8.75	8.95	11.03	14.62	19.37	23.50	25.29	23.95	20.76	16.37	12.48
2	COSMOS	7.91	6.68	6.38	7.34	9.53	12.23	14.95	15.94	14.94	13.34	11.49	9.58
3	EC-Earth3.3	9.18	7.62	7.41	9.32	13.58	19.22	24.06	25.27	23.56	20.26	16.05	12.13
4	GISS2.1G	4.87	3.58	3.88	6.75	11.21	15.56	18.68	19.39	17.68	14.47	10.53	7.18

	modelname	ja	fb	mr	ar	my	jn	jl	ag	sp	ot	nv	dc
0	CCSM4	8.39	8.81	9.63	12.05	15.15	19.61	22.81	22.96	19.02	14.70	11.03	9.13
1	CCSM4-UoT	6.97	7.58	9.96	12.54	16.46	21.92	25.43	24.72	19.69	14.32	9.94	7.02
2	CCSM4-Utr	9.43	9.50	10.88	13.14	16.81	21.09	25.02	24.99	21.79	17.26	12.84	10.15
3	CESM1.2	9.32	9.84	10.56	13.06	16.71	20.91	23.83	23.44	20.11	15.96	12.77	10.38
4	CESM2	10.70	10.44	11.75	13.85	17.69	22.31	25.02	25.18	21.79	17.65	13.99	12.05

month	modelname	ar	ag	dc	fb	ja	jl	jn	mr	my	nv	ot	sp
0	CCSM4-UoT	18.641620	17.664583	19.175005	19.155079	19.240961	17.706797	17.993721	18.922827	18.357751	18.862000	18.452913	17.934462
1	COSMOS	19.671519	19.139396	19.822553	19.930394	19.886921	19.228002	19.358863	19.860139	19.468221	19.729077	19.523501	19.264149
2	HadCM3	35.505535	34.605678	35.505248	35.616506	35.583428	34.878770	35.134441	35.604174	35.330172	35.355503	35.068206	34.696112
3	HadGEM3	32.947459	32.489019	32.896642	33.094949	33.066085	32.465359	32.599833	33.040556	32.797878	32.658298	32.534356	32.513514
4	IPSL-CM6A-LR	34.079274	33.265265	34.244052	34.275511	34.315632	33.524891	33.752462	34.183905	33.928167	34.027614	33.766878	33.409809

month	modelname	ar	ag	dc	fb	ja	jl	jn	mr	my	nv	ot	sp
0	CCSM4	1.481677	1.825992	4.034268	2.957591	3.776682	0.731645	0.996839	1.921807	1.343585	3.962916	3.387227	2.900726
1	CCSM4UOT	1.494104	1.732526	3.068386	2.104363	2.758223	0.965024	1.286421	1.389566	1.473732	3.428390	2.597931	2.323011
2	CCSM4Utr	1.525553	1.702673	3.965647	2.613341	3.236507	1.256546	1.676994	1.656356	1.695042	3.999845	2.977811	2.365357
3	CESM1.2	1.461625	2.224233	3.428275	2.466815	3.199764	0.845094	0.787780	2.065939	1.132148	3.458813	3.049558	2.940248
4	CESM2	1.549717	2.337191	4.229531	2.705796	3.424249	1.708422	1.124801	1.911965	1.448587	4.192045	3.278223	2.665053

	modelname	ja_SST	fb_SST	mr_SST	ar_SST	my_SST	jn_SST	jl_SST	ag_SST	sp_SST	...	dc_precip	fb_precip	ja_precip	jl_precip	jn_precip	mr_precip	my_precip	nv_precip	ot_precip	sp_precip
0	ccsm4	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	...	4.034268	2.957591	3.776682	0.731645	0.996839	1.921807	1.343585	3.962916	3.387227	2.900726
1	ccsm4uot	8.33	7.14	7.45	9.67	13.33	18.40	22.5	24.35	22.60	...	3.068386	2.104363	2.758223	0.965024	1.286421	1.389566	1.473732	3.428390	2.597931	2.323011
2	ccsm4utr	9.96	8.75	8.95	11.03	14.62	19.37	23.5	25.29	23.95	...	3.965647	2.613341	3.236507	1.256546	1.676994	1.656356	1.695042	3.999845	2.977811	2.365357
3	cesm1.2	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	...	3.428275	2.466815	3.199764	0.845094	0.787780	2.065939	1.132148	3.458813	3.049558	2.940248
4	cesm2	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	NaN	...	4.229531	2.705796	3.424249	1.708422	1.124801	1.911965	1.448587	4.192045	3.278223	2.665053

INTRO¶

Notebook for applying Data Assimilation on seasonal data (PWP reconstructions)¶

Characteristics:¶

Load packages¶

PRIOR - MONTHLY¶

Load monthly SST model data¶

Load monthly SAT model data¶

Load monthly SSS model data¶

Load monthly precipitation data¶

Combine SST, SAT, SSS and precipitation data by modelname¶

Calculate the monthly prior for model SST, SAT, SSS and precipitation values¶

Plot the monthly priors for all model values¶

Convert SST and SAT model data to D47 domain using the regression by Daëron and Vermeesch (2023) and propagate uncertainty in the calibration¶

Estimate seawater oxygen isotope value from salinity based on different salinity-d18Ow relationships:¶

Calculate carbonate oxygen isotope value from SST and seawater oxygen isotope data using Grossman and Ku (1986) with the VPDB-VSMOW scale correction by Gonfiantini et al. (1995) and Dettman et al. (1999)¶

Calculate the monthly prior for model SST- and SAT-derived D47 values and SSS-derived seawater oxygen isotope values with propagated uncertainty¶

Plot the monthly prior for model SST- and SAT-derived D47 values, model SSS-derived carbonate d18O values and precipitation with propagated uncertainty¶

Plot the monthly prior for model SSS-derived carbonate d18O values using different SSS-d18Ow relationships¶

Calculate the monthly covariance matrix for D47 values of SST and SAT, d18Oc and precipitation¶

Plot normalized monthly covariance matrix between D47 values of SST and SAT, d18Oc and precipitation¶

Create combined monthly state vector in D47 and d18Oc domain¶

OBSERVATIONS¶

Load clumped data for updating¶

Seasonal data from PWP paper¶

Monthly and seasonal data calculated from ShellChron outcomes¶

Prepare measurement and observation matrices¶

Define a wrapped normal distribution to allow uncertainty in the time domain to flow around the year¶

Observations on monthly scale¶

Measurement matrix for monthly aggregated D47 values¶

Observation matrix for monthly aggregated D47 data¶

DEFINE UPDATING FUNCTIONS¶

Create updating function (Kalman filter)¶

Create function to track the statistics of the likelihood (combining just the reconstruction data)¶

EXECUTE UPDATING FUNCTIONS - MONTHLY¶

Update monthly prior with data (d18Ow conversion following Harwood)¶

Plot monthly posterior without sclero-dating uncertainty in D47 domain¶

Update monthly prior with data (d18Ow conversion following Graniero)¶

Plot monthly posterior without sclero-dating uncertainty in D47 domain¶

Update monthly prior with data (d18Ow conversion following Railsbeck assuming icehouse conditions)¶

Plot monthly posterior without sclero-dating uncertainty in D47 domain¶

Update monthly prior with data (d18Ow conversion following assuming greenhouse conditions)¶

Plot monthly posterior without sclero-dating uncertainty in D47 domain¶

POSTERIOR - MONTHLY¶

Monthly posterior in temperature and salinity domains with aggregated data¶

Convert monthly prior, likelihood and posterior to temperature and salinity¶

Plot SAT, SST, SSS and precipitation posterior with SDs instead of 95% confidence levels using Harwood for d18Ow-SSS conversion¶

Plot SAT, SST, SSS and precipitation posterior with SDs instead of 95% confidence levels using Graniero for d18Ow-SSS conversion¶

Plot SAT, SST, SSS and precipitation posterior with SDs instead of 95% confidence levels using Railsback (icehouse) for d18Ow-SSS conversion¶

Plot SAT, SST, SSS and precipitation posterior with SDs instead of 95% confidence levels using Railsback (greenhouse) for d18Ow-SSS conversion¶

Plot SST estimates and precision difference between assimilations with different d18Ow-SSS relationships¶

Aggregate statistics on posterior outcomes on SST-D47 and d18Oc for all cases¶

Update aggregate table to innovation values by subtracting SST priors from SST posteriors¶

Create aggregate table containing SST priors and posteriors organized by SSS-d18Ow method¶

Plot monthly SST innovation outcomes and uncertainties per d18Ow-SSS relationship¶

Plot monthly SST outcomes and uncertainties per d18Ow-SSS relationship¶